Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Integer solutions to $x^3+y^3+z^3 = x+y+z = 8$
Find all integers $x,y,z$ that satisfy $$x^3+y^3+z^3 = x+y+z = 8$$
Let $a = y+z, b = x+z, c = x+y$. Then $8 = x^3+y^3+z^3 = (x+y+z)^3-3abc$ and therefore $abc = 168$ and $a+b+c = 16$. Then do I just use the prime factorization of $168$?
| Hint:
Taking from where you left off: $ab \mid 168 \implies ab = \pm 1, \pm 2, \pm 4, \pm 6, \pm 7, \pm 8, \pm 12, \pm 14, \pm 21, \pm 24, \pm 28, \pm 42, \pm 56, \pm 84, \pm 168$. Even though it looks cumbersome, it is easy to solve. For example, $ab = 6 \implies c = \dfrac{168}{6} = 28 \implies a+b = 16-c = 16 - 28 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1870805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Algebraic solution for the value of $x$. I solved this problem the fifteen years ago without numerically solving equations of degree 4, I was happy in a substitution that I avoid directly attacking equations of degree 4.
Today my nephew, who is an enthusiastic student of mathematics, proposes me the same problem. It ... | From $y = 1/x,$ then multiplying by $x^2,$ i got
$$ x^4 + 2 x^3 - x^2 - 2 x - 1. $$
This looks bad. However, set
$$ x = t - \frac{1}{2} $$
and you get rid of the cubic term, always worth a try. I was pleased to discover that the linear term also vanished, giving
$$ t^4 - \frac{5}{2} t^2 - \frac{7}{16}, $$
and you can s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1871168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Joint probability density function $(X^2,Y^2)$ Let $X$ and $Y$ be random variables having the following joint probability density function
$f(x,y)=\begin{cases}
\frac{3}{8}xy & x\geq0,\,y\geq0,\:x+y\leq2,\\
0 & \mbox{otherwise}.
\end{cases}$
Find the joint probability density function of $X^2$ and $Y^2$.
This is my... | I think the issue is the original pdf:
$$ \int_{\mathbb{R}^2}f(x,y)\;dydx=\frac{3}{8}\int_0^2\int_{0}^{2-x}xy\;dydx=\frac{3}{8}\int_0^2\frac{x(2-x)^2}{2}\;dx$$
$$ =\frac{3}{8}\int_0^2\frac{(2-x)x^2}{2}\;dx=\frac{3}{8}\cdot\frac{2}{3}=\frac{1}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1873478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integral over $0\leq x,y,z\leq 1$ and $x+y+z\leq 2$ What is
$$\int_{S}(x+y+z)dS,$$ where $S$ is the region $0\leq x,y,z\leq 1$ and $x+y+z\leq 2$?
We can change the region to $0\leq x,y,z\leq 1$ and $x+y+z\geq 2$, because the total of the two integrals is just
$$\int_0^1\int_0^1\int_0^1(x+y+z)dxdydz=3\int_0^1xdxdydz=\fr... | You could write
\begin{equation}
\int_S x \, dx dy dz = \int_0^1 \left( \int_{y+z \leq 2-x; \, 0\leq y,z \leq 1} dy dz \right) x dx
\end{equation}
Now, you can interpret $y+z \leq 2-x$ with $y,z \geq 0$ as a triangle in the plane, whose area is $\frac{(2-x)^2}{2}$.
From this triangle, you subtract two smaller triangle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1876732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Spivak Calculus - Chapter 1 Question 4.6 In Spivak's Calculus, Chapter 1 Question 4.6:
Find all the numbers $x$ for which $x^2+x+1>2$
The chapter focuses on using the following properties of numbers to prove solutions are correct:
Based on those properties, I am able to perform the following algebra:
$
\begin{align... | Complete the Square
$
\begin{align}
x^2+x+1&>2 & \text{Given}\\
x^2+x+1+0&>2+0 & \text{By Addition}\\
x^2+x+1+0&>2 & \text{By P2}\\
x^2+x+0+1&>2 & \text{By P4}\\
x^2+x+\left( \frac{1}{2} \right)^2+(-1)\left( \frac{1}{2} \right)^2+1 &>2 & \text{By P3}\\
\left(x+\frac{1}{2}\right)\left(x+\frac{1}{2}\right)+(-1)\left( \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1878298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Solve $\int_{0}^{1}\frac{1}{1+x^6} dx$ Let $$x^3 = \tan y\ \ \text{ so that }\ x^2 = \tan^{2/3}y$$
$$3x^2dx = \sec^2(y)dy$$
$$\int_{0}^{1}\frac{1}{1+x^6}dx = \int_{1}^{\pi/4}\frac{1}{1+\tan^2y}\cdot \frac{\sec^2y}{3\tan^{2/3}y}dy = \frac{1}{3}\int_{1}^{\pi/4} \cot^{2/3}y\ dy$$
How should I proceed after this?
EDITED: C... | Continuing where MK12 left off, we proceed as follows:
$$\frac{2-x^2}{x^4-x^2+1} = \frac{1}{2} \times \frac{4-2x^2}{x^4-x^2+1} = \frac{1}{2} \times \frac{1+x^2 + 3(1-x^2)}{x^4-x^2+1}$$
Split the sub-integral into two parts. For one, make the substitution $u=x+\frac{1}{x}$ and the other $v = x-\frac{1}{x}$
Then the rest... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1882650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Solution of $ydx-xdy+3x^2y^2e^{x^2}dx=0$ Find the solution of given differential equation:
$$ydx-xdy+3x^2y^2e^{x^2}dx=0$$
I am not able to solve this because of $e^{x^2}$. Could someone help me with this one?
| By dividing both side to $y^2$ we get $$ydx-xdy+3x^{ 2 }y^{ 2 }e^{ x^{ 2 } }dx=0\\ \frac { ydx-xdy }{ { y }^{ 2 } } +3x^{ 2 }e^{ x^{ 2 } }=0\\ d\left( \frac { x }{ y } \right) =3x^{ 2 }e^{ x^{ 2 } }\\ \int { d\left( \frac { x }{ y } \right) =\int { 3x^{ 2 }e^{ x^{ 2 } }dx } } =\frac { 3 }{ 2 } \int { x } d{ e }^{ { ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1888503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Evaluate the reciprocal of the following infinite product I hae to evaluate the reciprocal of the following product to infinity
$$\frac{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9 \cdot 11 \cdot 13 \cdot 15 \cdot 17 \cdot 19}{2 \cdot 2 \cdot 6 \cdot 6 \cdot 10 \cdot 10 \cdot 14 \cdot 14 \cdot 18 \cdot 18}\cdot\ldots $$
I am gues... | Note that $$1-\frac{1}{\left(4n-2\right)^{2}}=\frac{\left(4n-3\right)\left(4n-1\right)}{\left(4n-2\right)\left(4n-2\right)}
$$ and $$P=\prod_{n\geq1}\left(1-\frac{1}{\left(4n-2\right)^{2}}\right)^{-1}=\prod_{n\geq1}\frac{\left(4n-2\right)\left(4n-2\right)}{\left(4n-3\right)\left(4n-1\right)}
$$ $$=\prod_{n\geq0}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1890127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solve $2\ddot{y}y - 3(\dot{y})^2 + 8x^2 = 0$ Solve differential equation
$$2\ddot{y}y - 3(\dot{y})^2 + 8x^2 = 0$$
I know that we have to use some smart substitution here, so that the equation becomes linear.
The only thing I came up with is a smart guessed particular solution: $y = x^2$. If we plug this function in, w... | Hint:
Let $y=\dfrac{1}{u^2}$ ,
Then $y'=-\dfrac{2u'}{u^3}$
$y''=\dfrac{6(u')^2}{u^4}-\dfrac{2u''}{u^3}$
$\therefore\dfrac{2}{u^2}\left(\dfrac{6(u')^2}{u^4}-\dfrac{2u''}{u^3}\right)-3\left(-\dfrac{2u'}{u^3}\right)^2+8x^2=0$
$\dfrac{12(u')^2}{u^6}-\dfrac{4u''}{u^5}-\dfrac{12(u')^2}{u^6}=-8x^2$
$\dfrac{4u''}{u^5}=8x^2$
$u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1891382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Stuck in integration: $\int {\frac{dx}{( 1+\sqrt {x})\sqrt{(x-{x}^2)}}}$
$\displaystyle\int {\frac{dx}{( 1+\sqrt {x})\sqrt{(x-{x}^2)}}}$
$=\displaystyle\int\frac{(1-\sqrt x)}{(1+x)\sqrt{x-x^2}}\,dx$
$=\displaystyle\int\frac{(1-\sqrt x+x-x)}{(1+x)\sqrt{x-x^2}}\,dx$
$=\displaystyle\int\frac{\,dx}{\sqrt{x-x^2}}-\disp... | By subtsitution twice we get $t=\sqrt { x } \Rightarrow dt=\frac { dx }{ 2\sqrt { x } } $ $$\int { \frac { dx }{ \left( 1+\sqrt { x } \right) \sqrt { x-{ x }^{ 2 } } } } =2\int { \frac { d\sqrt { x } }{ \left( 1+\sqrt { x } \right) \sqrt { 1-x } } = } 2\int { \frac { dt }{ \left( 1+t \right) \sqrt { 1-{ t }^{ 2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1892802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Help solving $1 < \frac{x + 3}{x - 2} < 2$ I worked a lot of inequalities here in MSE and that greatly helped me.
I've seen a similar inequality [here] [1], but the one I have today is significantly different, in that I'll end up with a division by 0, which is not possible.
[1] [Simple inequality
$$1 < \frac{x + 3}{x -... | Break it into two separate inequalities when you get to here:
$$ 0 < \frac{5}{x-2} < 1$$
First let's consider $0 < \dfrac{5}{x-2}$. Since $5 > 0$, this inequality is satisfied when $x-2 > 0$, i.e., when $x > 2$.
Now let's consider $\dfrac{5}{x-2} < 1$. Subtract $1$ from both sides and get a common denominator to get ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1892918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Find the fourier series representation of a function
Consider the function
$f(x) =
\begin{cases}
\frac{\pi}{2}+x & & x \in (-\pi, 0] \\
\frac{\pi}{2}-x & & x \in (0, \pi]\\
\end{cases}$
extended 2$\pi$ periodically to $\mathbb{R}$. Calculate $a_0, a_n, b_n$
I understand how to work out a fourier series but I am ... | Divide it two parts and calculate $$a_{ 0 }=\frac { 1 }{ \pi } \int _{ -\pi }^{ \pi } f\left( x \right) dx=\frac { 1 }{ \pi } \int _{ -\pi }^{ 0 }{ \left( \frac { \pi }{ 2 } +x \right) dx } +\frac { 1 }{ \pi } \int _{ 0 }^{ \pi }{ \left( \frac { \pi }{ 2 } -x \right) dx } =\\ ={ \left( \frac { \pi }{ 2 } x+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1893008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Inequality on non-negative reals For non-negative $x,y,z$ satisfy $\frac{1}{2x^2+1}+\frac{1}{2y^2+1}+\frac{1}{2z^2+1}=1$ then show that $x^2+y^2+z^2+6\geq 3(x+y+z)$
Idea how to handle the constraint? I'm unaware .
| Yes, it's true for all reals.
$\sum\limits_{cyc}(x^2-3x+2)=\sum\limits_{cyc}\left(x^2-3x+2-\frac{9}{4}\left(\frac{1}{2x^2+1}-\frac{1}{3}\right)\right)=\sum\limits_{cyc}\frac{(x-1)^2(2x-1)^2}{2(2x^2+1)}\geq0$.
Done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1893184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Two lines through a point, tangent to a curve We are looking for two lines through $(2,8)$ tangent to $y=x^3$. Let's denote the intersection point as $(a, a^3)$ and use the slope equation together with the derivative to get $\frac{a^3-8}{a-2}=3a^2$. This yields a cubic equation. Of course, one of the lines is tangent t... | The tangents must be of the form
$$y-8=m(x-2),$$ and they intersect the cubic $\color{blue}{y=x^3}$ when
$$x^3-8=m(x-2).$$
This equation must have a double root, so that differentiating on $x$, we also have
$$3x^2=m.$$
With the obvious solution $x=2$, we deduce $m=12$ and
$$\color{green}{y-8=12(x-2)}.$$
Otherwise, we m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1895466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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How would I write a function for the following pattern? \begin{align}
Y(0) ={}& 1\\
Y(1) ={}& 2.5\\
Y(2) ={}& 2.5\cdot2.3\\
Y(3) ={}& 2.5\cdot 2.3\cdot 2.1\\
Y(4) ={}& 2.5\cdot 2.3\cdot 2.1\cdot 1.9\\
\vdots\,\,\,
\end{align}
How would I solve for something like $Y(1.3)$ or $Y(2.7)$? How would a function for $Y(x)$ be ... | Notice that
\begin{align*}
Y(x) &= \frac{25}{10} \cdot \frac{23}{10} \cdot \frac{21}{10} \cdots \frac{27 - 2x}{10} \\[5pt]
&= \frac{1}{10^x} \cdot \frac{(26)(25)(24)(23)(22)(21) \cdots (28-2x)(27-2x)}{(26)(24)(22)\cdots (28-2x)} \\
&= \frac{1}{10^x} \cdot
\frac{(26)(25)(24)(23)(22)(21) \cdots (2)(1)}{(26)(24)(22)\cdots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1895551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $(a + b\sqrt c)^n = d + e\sqrt c$, then $(a - b\sqrt c)^n = d - e\sqrt c$ I think that:
If $a,b,(c\ge0$ not a prefect square$),d,e,f\in\mathbb Z$ such that for some $n \ge 1$, $(a + b\sqrt c)^n = d + e\sqrt c$, then $(a - b\sqrt c)^n = d - e\sqrt c$
Is this true? Can someone provide a proof or give a hint for how ... | Consider the set
\begin{align*}
A = \{ x + y\sqrt{c}: x, y \in \mathbb{Z} \}
\end{align*}
Consider the map $f: A \rightarrow A$ defined by $f(x+y\sqrt{c}) = x - y \sqrt{c}$. It is easy to see that $f$ is well defined (since $c$ is not a perfect square) and for any two $a_1, a_2 \in A$,
\begin{align*}
f(a_1+a_2) &= f(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Suppose we have two dice one fair and one tha brings $6$ with quintuple probability.Find the probability to throw randomly one die and show $6$ Suppose we have two dice one fair and one that brings $6$ with quintuple probability than the other numbers.We get a die randomly and we throw it.What is the probability to hav... | The probability of rolling 6 on the fair die is obviously $\frac{1}{6}$.
Let $x$ denote the probability of rolling 6 on the non-fair die:
*
*Then $\frac{1}{5}x$ is the probability of rolling each one of the other $5$ values
*Therefore $x+5\cdot\frac{1}{5}x=1$, therefore $2x=1$, therefore $x=\frac{1}{2}$
So the pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Squaring Infinite Series Expansion Of e^x $Fact$:$$\lim\limits_{n \to \infty}\frac{x^0}{0!}+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^n}{n!}=e^x$$
so
$$\lim\limits_{n \to \infty}\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\dots+\frac{1}{n!}=e$$
also
$$\lim\limits_{n \to \infty}e^2=\frac{2^0}{0!}+... | Since $e^ne^m = e^{n+m}$ you can simply let $x = 2t$, giving you
$$(e^{t})^2 = e^{2t} = 1 + 2t + \frac{4t^2}{2!}+\frac{8t^3}{3!} + \cdots$$
Now, let $t = 1$ and you have
$$e^2 = 1 + 2 + \frac{4}{2!} + \frac{8}{3!} + \cdots = (1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots )^2 = (e^1)^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Find the second smallest integer such that its square's last two digits are $ 44 $ Given that the last two digits of $ 12^2 = 144 $ are $ 44, $ find the next integer that have this property.
My approach is two solve the equation $ n^2 \equiv 44 \pmod{100}, $ but I do not know how to proceed to solve that equation.
I t... | If $x^2$ ends with $44$ then $x$ is even. Let $y=2x$. We are trying to solve
$$(2y)^2\equiv 44\pmod{100}$$
and this equation is equivalent to
$$y^2\equiv 11\pmod{25}$$
Since $6^2\equiv 11\pmod{25}$ this equation can be written as
$$(y-6)(y+6)\equiv 0\pmod{25}$$
It is not possible that both $y-6$ and $y+6$ are multiples... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Finding $a$ in quadratic equation $2x^2 - (a+1)x + (a-1)=0$ so that difference of two roots is equal to its product Given equation:
$$2x^2 - (a+1)x + (a-1)=0$$
I have to find when the difference of two roots is equal to its product, i.e.:
$$x_1x_2 = x_1 - x_2.$$
From Vieta's formulas we know that:
$$x_1 + x_2 = \frac{a... | Using by the formula $${ \left( { x }_{ 1 }-{ x }_{ 2 } \right) }^{ 2 }+4{ x }_{ 1 }{ x }_{ 2 }={ \left( { x }_{ 1 }+{ x }_{ 2 } \right) }^{ 2 }$$ make be easy
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Is the difference of two irrationals which are each contained under a single square root irrational? Is $ x^\frac{1}{3} - y^\frac{1}{3}$ irrational, given that both $x$ and $y$ are not perfect cubes, are distinct and are integers (i.e. the two cube roots are yield irrational answers)?
I understand that the sum/differen... | Denote the cube roots by $X,Y$, so that $X^3=x$ and $Y^3=y$ with $x,y\in \mathbb Z$.
Suppose, with slightly greater generality, that we have $X-Y-R=0$ where $X^3,Y^3,R\in \mathbb Q$.
We first want to argue that $XY\in \mathbb Q$. To do so, observe the identity:
$$X^3-Y^3-R^3-3XYR=(X-Y-R)(X^2+Y^2+R^2+XY+XR-YR)$$
This i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Evaluate the integral $\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$ This integral looks a lot like an elliptic integral, but with cubes instead of squares:
$$I(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$$
Let's consider $a,b>0$ for now.
$$I(a,a)=\int_0^\infty \frac{dx}{x^3+a^3}=\frac{2 \pi}{3 \sqrt{... | More generally, with $|p-1|<1$, some experimentation shows that,
$$\int_0^\infty \frac{dt}{\sqrt{(t^m+1)(t^m+p)}} = \pi\,\frac{\,_2F_1\big(\tfrac12,\tfrac{m-1}{m};1;1-p\big)}{m\sin\big(\tfrac{\pi}{m}\big)}$$
where the question was just the case $m=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 1
} |
Multi-index sum property Exercise 1.2.3.29 in Donald Knuth's The Art of Computer Programming (3e) states the following property of a multi-indexed sum:
$$
\sum_{i=0}^n \sum_{j=0}^i \sum_{k=0}^j a_ia_ja_k = \frac{1}{3}S_3 + \frac{1}{2}S_1S_2 + \frac{1}{6}S_1^3,
$$
where $S_r = \sum_{i=0}^n a_i^r$.
I tried to prove it an... | For future reference this is the sum over all multisets of size three
chosen from the variables $A_0$ to $A_n$ and evaluated at $a_q.$
Therefore by the Polya Enumeration Theorem it is given by
$$\left.Z(S_3)\left(\sum_{q=0}^n A_q\right)\right|_{A_q=a_q}$$
where $Z(S_3)$ is the cycle index of the symmetric group ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Find the value of $\tan A + \tan B$, given values of $\frac{\sin (A)}{\sin (B)}$ and $\frac{\cos (A)}{\cos (B)}$ Given
$$\frac{\sin (A)}{\sin (B)} = \frac{\sqrt{3}}{2}$$
$$\frac{\cos (A)}{\cos (B)} = \frac{\sqrt{5}}{3}$$
Find $\tan A + \tan B$.
Approach
Dividing the equations, we get the relation between $\tan A$ and $... | Although there is something wrong with this question,there is a way which I think maybe a little bit easier to solve this kind of problem.
$$\tan A + \tan B = \frac{\sin(A)\cdot\cos(B) + \sin(B)\cdot\cos(A)}{\cos(A)\cdot\cos(B)} = \frac{{\sin A \over \sin B}+ {\cos A \over \cos B}}{\cos A \over \sin B} $$
then to ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Product of roots of $ax^2 + (a+3)x + a-3 = 0$ when these are positive integers There is only one real value of $'a'$ for which the quadratic equation $$ax^2 + (a+3)x + a-3 = 0$$ has two positive integral solutions.The product of these two solutions is :
Since the solutions are positive, therefore the product of roots ... | We need $$\frac{a+3}{a}\in\Bbb{Z}\ , \frac{a-3}{a}\in\Bbb{Z}$$ or $$1+\frac{3}{a}\in\Bbb{Z}\ , \ 1-\frac{3}{a}\in\Bbb{Z}$$ thus $\displaystyle \frac{3}{a}\in\Bbb{Z}$, means that $\displaystyle a=\frac{3}{m}$ where $m\in\Bbb{Z}$.
Now we can write the equation as $$\frac{3}{m}x^2+\left(\frac{3}{m}+3\right)x+\frac{3}{m}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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If $a$, $b$, and $c$ are sides of a triangle, then $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$.
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that
$$\sum_{\text{cyc}}\frac{a}{b+c}=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2\,.$$
Attempt. By clearing the denominators, the required inequality is equiv... | \begin{align*}
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} & = \frac{2a}{2(b+c)}+\frac{2b}{2(c+a)}+\frac{2c}{2(a+b)} \\
&< \frac{2a}{a+b+c} + \frac{2b}{c+a+b} + \frac{2c}{a+b+c} \\
&= 2
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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Is A276175 integer-only? The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}a_{n-1}-a_{n-2}-1$).
But is it also true for the sequence A276175 defined by $a_0=a_1=a_2=a_3=... | Yes, $(a_n)$ is a sequence of integers.
To prove this we first need to study some auxiliary sequences that satisfy a polynomial recurrence relation (unlike $(a_n)$ which has a rational fraction as its recurrence).
Consider the sequences $(b_n)$ of positive reals satisfying the recurrence relation $b_nb_{n+4} = b_{n+1}b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 1,
"answer_id": 0
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Trouble understanding proof of the inequality - $(\frac{1}{a}+1)(\frac{1}{b}+1)(\frac{1}{c}+1) \ge 64 $, for $a,b,c > 0$ and $a+b+c = 1$ I was looking into this problem in a book discussing inequalities, However I found the proof quite hard to understand.The problem is as follows:
Let $a,b,c$ be positive numbers with ... | using for $$\frac{1}{a},\frac{1}{b},\frac{1}{c}$$ the AM-GM inequality we obtain
$$\frac{1}{3}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq \sqrt[3]{\frac{1}{abc}}$$ and for $$\frac{1}{ab},\frac{1}{bc},\frac{1}{ca}$$ the same we get $(3)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Prove $\int_0^\infty \frac{dx}{\sqrt{(x^4+a^4)(x^4+b^4)}}=\frac{\pi}{2 \sqrt2 a b} ( \text{agm} (\frac{a+b}{2},\sqrt{\frac{a^2+b^2}{2}} ))^{-1}$ The following definite integral turns out to be expressible as the Arithmetic-Geometric Mean: $$I_4(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^4+a^4)(x^4+b^4)}}=\frac{\pi}{2 \sqrt2... | Substitition is sufficient.
Let $$\displaystyle z=x-\frac1x,w=x+\frac1x$$
then $$\displaystyle
\frac{\mathrm dx}{\sqrt{x^8+p x^4+1}}=\frac12\left(
\frac{\mathrm dz}{\sqrt{z^4+4z^2+2+p}}+\frac{\mathrm dw}{\sqrt{w^4-4w^2+2+p}}\right)
$$
So $$f(p)=\displaystyle
\int_0^\infty
\frac{\mathrm dx}{\sqrt{x^8+p x^4+1}}
=\frac12\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1905349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
If $a+b+c=0$ then the roots of $ax^2+bx+c=0$ are rational? If $a+b+c=0$ then the roots of $ax^2+bx+c=0$ are rational ?
Is it a "If and only if " statement or "only if " statement ?
For $a,b,c \in \mathbb Q$ , I think it is a "if and only if" statement . Am I correct ?
I can prove that if $a+b+c=0$ and $a,b,c \in \mathb... | If $c = -a-b$, then the discriminant is $$b^2-4ac = b^2 +4a(a+b) = b^2 +4ab +4a^2 = (b+2a)^2.$$
Since the discriminant is a perfect square, then the roots are always rational.
Their values are
$$\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}
=\dfrac{-b \pm (b+2a)}{2a}
\in \left\{ 1, -\dfrac{a+b}{a} \right\}$$
Of course, now ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1906065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Evaluate the integral $\int_0^\pi \sin{(x \cos{t}})\cos{t}\; dt$ How to evaluate:
$\int \sin{(x \cos{t}})\cos{t}\; dt$
or:
$\int_0^\pi \sin{(x \cos{t}})\cos{t}\; dt$
| $\int\sin(x\cos t)\cos t~dt=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}\cos^{2n+2}t}{(2n+1)!}~dt$
For $n$ is any non-negative integer,
$\int\cos^{2n+2}t~dt=\dfrac{(2n+2)!t}{4^{n+1}((n+1)!)^2}+\sum\limits_{k=0}^n\dfrac{(2n+2)!(k!)^2\sin t\cos^{2k+1}t}{4^{n-k+1}((n+1)!)^2(2k+1)!}+C$
This result can be done by succe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1907526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Show that any integer $n>7$ can be written as the sum of $3$s and $5$s exclusively
Show that any integer $n>7$ can be written as the sum of $3$s and $5$s exclusively,
i.e.,
$$
8= 5+3 \\
9=3+3+3 \\
10 = 5+5 \\
11 = 5+3+3 \\
12 = 3+3+3+3
$$
So I've started in a couple directions without progress. I think it makes ... | $2*3 - 5 = 1$
$2n*3 - 5n = n$.
$3(2n - 5k) + 5(3k - n)= n$.
To assure that $2n - 5k \ge 0$ and $3k - n > 0$...
If $n = 3m - r; r = 0, 1,2$ then $k$ can be anything equal or greater than $m$ so long as $2n - 5k \ge 0$ i.e. $6m - 2r - 5k \ge 0\implies k \le 6m/5 - 2r/5= m + \frac{m-2r}5$.
So long as $m \ge 4$ we will a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
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Counting using permutation and combination How many solutions are there to the equation
$x_1+x_2+x_3+x_4+x_5=21$,
where $x_i,i=1,2,3,4,5$, is a nonnegative integer such
that
$ 0 ≤ x_1 ≤ 3$, $1 ≤ x_2 < 4$, and $x_3 ≥ 15$?
I tried it .My Approach-:
$ x_3=x_3'+15 \implies x_1+x_2+x_3'+15+x_4+x_5=21 \implies x_1+x_2+x_3... | Let $y_3 = x_3 -15$. We need the number of solutions to
$x_1+x_2+y_3+x_4+x_5 = 21 - 15$ with $y_3 \geq 0$, $0 \leq x_1 \leq 3$ and $1 \leq x_2 <4$. The number of solutions is the coefficient of $x^6$ in
\begin{align*}
(1+x+x^2+x^3)&(x+x^2+x^3)(1+x+x^2+\cdots)(1+x+x^2+\cdots)(1+x+x^2+\cdots)\\
&= x(1+x+x^2+x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Show $A^T$ has an eigenvector with all components rational
Matrix $A$ is a $5 \times 5$ matrix with rational entries such that $(1, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5})^T$ is an eigenvector of A. Show that $A^T$ has eigenvector with all components rational.
My idea is: let the eigenvalue associated with the above ... | Let $v = (1, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5})^T$ and assume $A v = \lambda v$. From the first row we get
$$a_{1,1} + 2 a_{1,4} + a_{1,2} \sqrt{2} + a_{1,3} \sqrt{3} + a_{1,5} \sqrt{5} = \lambda$$
From the second row we get:
$$a_{2,1} + 2 a_{2,4} + a_{2,2} \sqrt{2} + a_{2,3} \sqrt{3} + a_{2,5} \sqrt{5} = \lambda ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1909766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How would you solve this polynomial? Is there a way to find the roots of equations such as $x^3-9\sqrt[3]{2}+9=0$?
I've just been using Wolfram Alpha to factor it into $(x-\sqrt[3]{4}+\sqrt[3]{2}-1)(x^2+(1-\sqrt[3]{2}+\sqrt[3]{4})x+3\sqrt[3]{4}-3)$. But for harder equations such as $$x^3-63\sqrt[3]{20}+9=0$$, Wolfram A... | We can find $\sqrt[3]{9(\sqrt[3]2-1)}$ by the following way without WA.
Indeed, let $\sqrt[3]2=x$.
Hence, $$x^3=2$$ or $$9(x^3-1)=9$$ or $$9(x-1)=\frac{9}{1+x+x^2}$$ or $$9(x-1)=\frac{27}{x^3+3x^2+3x+1}$$ or
$$\sqrt[3]{9(\sqrt[3]2-1)}=\frac{3}{\sqrt[3]2+1}$$ or
$$\sqrt[3]{9(\sqrt[3]2-1)}=\sqrt[3]4-\sqrt[3]2+1$$
and we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1913452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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The integral $\int\frac{2(2y^2+1)}{(y^2+1)^{0.5}} dy$ What is $$\int\frac{2(2y^2+1)}{(y^2+1)^{0.5}} dy?$$ I split it as $\frac{y^{2}}{(y^2+1)^{0.5}} + \sqrt{y^2+1}.$ Now I substituted $y^{2}=u $ thus $2y\,dy=du$ so we get $0.5 \sqrt{\frac{u}{u + 1}} + 0.5 \sqrt{\frac{1 + u}{u}}$ but now what to do? Another idea was doi... | $\displaystyle\int\frac{4y^2+2}{\sqrt{y^2+1}}dy=\int\frac{2y^2+2}{\sqrt{y^2+1}}dy+\int\frac{2y^2}{\sqrt{y^2+1}}dy=2\int\sqrt{y^2+1}dy+\int\frac{2y^2}{\sqrt{y^2+1}}dy.$
Now use $\displaystyle u=2y,\;dv=\frac{y}{\sqrt{y^2+1}}dy\;$ so $\;du=2dy,\;v=\sqrt{y^2+1}$ in the 2nd integral to obtain
$\displaystyle2\int\sqrt{y^2+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1914515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Prove: $\cos^3{A} + \cos^3{(120°+A)} + \cos^3{(240°+A)}=\frac {3}{4} \cos{3A}$ Prove that:
$$\cos^3{A} + \cos^3{(120°+A)} + \cos^3{(240°+A)}=\frac {3}{4} \cos{3A}$$
My Approach:
$$\mathrm{R.H.S.}=\frac {3}{4} \cos{3A}$$
$$=\frac {3}{4} (4 \cos^3{A}-3\cos{A})$$
$$=\frac {12\cos^3{A} - 9\cos{A}}{4}$$
Now, please help m... | use that $$\cos(120^{\circ}+x)=-1/2\,\cos \left( x \right) -1/2\,\sqrt {3}\sin \left( x \right) $$
and $$\cos(240^{\circ}+x)=-1/2\,\cos \left( x \right) +1/2\,\sqrt {3}\sin \left( x \right) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1921191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 8,
"answer_id": 4
} |
Proof of quadratic inequality using AM-GM Proof of quadratic inequality using AM-GM
| we have to show that $$x^2y^2+x^2+y^2+4-6xy\geq 0$$ dividing by $x^2+1$ gives
$$y^2-\frac{6xy}{x^2+1}+\frac{4+x^2}{x^2+1}\geq 0$$ this is equivalent to
$$\left(y-\frac{3x}{x^2+1}\right)^2+\frac{(4+x^2)(x^2+1)-9x^2}{(x^2+1)^2}\geq 0$$
and this is equivalent to $$\left(y-\frac{3x}{x^2+1}\right)^{ 2 }+\frac{(x^2-2)^2}{(x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1922395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Let A be an $n*n$ matrix. Prove that if $rank(A) = 1$, then $det(A + E) = 1 + trace(A)$ I feel like I've got the answer, but I've never been good at putting what I think into words.
$\begin{vmatrix}
n_{11} & n_{12} \\
n_{21} & n_{22}
\end{vmatrix} = 0 = n_{11}n_{22} - n_{12}n_{21}$
$\begin{vmatrix}
n_{11} + 1 & n_{12} ... | The value of a determinant as well of a trace is independent of the choice of basis. So suppose that the image of $A$ is generated by a vector $v_1$. Complement this vector with $v_2,...,v_n$ to form a base. In this base the matrix of $A$ takes the form:
$$ \underline{A} =
\left( \begin{matrix} a_{11} & a_{12} & ..... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to factorize $5$ in $\mathbb{Z}[\root 3 \of 2]$? Since $5$ has a norm of $125$ in this domain, and $N(1 + (\root 3 \of 2)^2) = 5$, it seems like a sensible proposition that $5 = (1 + (\root 3 \of 2)^2) \pi_2 \pi_3$, where $\pi_2, \pi_3$ are two other numbers in this domain having norms of $5$ or $-5$. This is suppo... | We have
$$ \mathbf Z[\sqrt[3]{2}]/(5) \cong \mathbf Z[x]/(x^3 - 2, 5) \cong \mathbf Z_5[x]/(x^3 - 2) \cong \mathbf Z_5[x]/(x+2) \times \mathbf Z_5[x]/(x^2 + 3x + 4) $$
so that the ideal $ (5) $ factors as $ (5) = \mathfrak p_1 \mathfrak p_2 $. To find the ideals $ \mathfrak p_1 $ and $ \mathfrak p_2 $, note that they c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1924378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
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Cayley graph interpretation D3 I am trying to understand the Cayley graph for the group $D_3$, which from Mathematica, I got:
I tried to get the multiplication table in Mathematica:
$\left(
\begin{array}{cccccc}
1 & 2 & 3 & 4 & 5 & 6 \\
2 & 1 & 4 & 3 & 6 & 5 \\
3 & 5 & 1 & 6 & 2 & 4 \\
4 & 6 & 2 & 5 & 1 & 3 \\
5 ... | The table and graph are related as follows. Let $X=\{1,2,3,4,5,6\}$ and let $S=\{2,4\}$. The table is missing its "headings" so that the actual table looks like this
$$
\begin{array}{c|cccccc}
& 1 & 2& 3& 4& 5&6\\
\hline
1 & 1 & 2 & 3 & 4 & 5 & 6 \\
2 & 2 & 1 & 4 & 3 & 6 & 5 \\
3 & 3 & 5 & 1 & 6 & 2 & 4 \\
4 & 4 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find $\int\limits^{\infty}_{0}\frac{1}{(x^8+5x^6+14x^4+5x^2+1)^4}dx$ I was asked to prove that
$$\int\limits^{\infty}_{0}\frac{1}{(x^8+5x^6+14x^4+5x^2+1)^{4}}dx=\pi\frac{14325195794+(2815367209\sqrt{26})}{14623232(9+2\sqrt{26})^\frac{7}{2}}$$
I checked the result numerically and the first digits correct using W|F
$$\in... | Hint. A route.
One may recall the following result, which goes back at least to G. Boole (1857).
Proposition. Let $f \in L^1(\mathbb{R})$ and let $f$ be an even function. Then
$$
\int_{-\infty}^{+\infty}x^{2n}f\left(x-\frac1x\right) dx=\sum_{k=0}^n \frac{(n+k)!}{(2k)!(n-k)!}\int_{-\infty}^{+\infty} x^{2k}f(x)\: dx. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
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Proving $\angle QAP=45^\circ$ if $ABCD$ is a square with points $P$ in $BC$, $Q$ in $CD$ satisfying $\overline{BP}+\overline{DQ}=\overline{PQ}$ Here is the problem:
Let $ABCD$ be a square with points $P$ in $BC$, $Q$ in $CD$ satisfying $\overline{BP}+\overline{DQ}=\overline{PQ}$. Prove that $\angle QAP=45^\circ$.
So ... | Using the cosine rule:$$|AP|^2+|AQ|^2-2|AP||AQ|cos\alpha=|PQ|^2$$Replacing $|AP|,|AQ|$ and $|PQ|$:$$|AP|=\sqrt{a^2+r^2}$$$$|PQ|=\sqrt{b^2+r^2}$$$$|PQ|=a+b$$
Here is $a=|BP|, b=|DQ|$:$$a^2+r^2+b^2+r^2-2\sqrt{a^2+r^2}\sqrt{b^2+r^2}cos\alpha=(a+b)^2$$$$\implies2r^2-2\sqrt{a^2+r^2}\sqrt{b^2+r^2}cos\alpha=2ab$$Note that $b ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1925585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Can completing the square apply to more higher degree? I have some trouble about completing the square lessons.
As we know , minimum value of $\ \ 3x^{2} + 4x + 1 $ is $-\frac{1}{3}$ when $x=-\frac{2}{3}$ by completing the square.
Then , how about minimum value of $\ \ x^5 +4x^4 + 3x^3 +2x^2 + x + 1$ when $(-2<x<2... | Yes we have something same. But if we look to the topic like following.
$3x^2+4x+1$ has a complete square $3x^2+4x+\frac43$ corresponding to it in which their difference, $-\frac13$, is of degree maximum two degrees less than the original.
For any polynomial in one variable we have a unique multiple of a power of a $(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1927629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Convergence/divergence of $\sum_{k=1}^\infty\frac{2\times 4\times 6\times\cdots\times(2k)}{1\times 3\times 5\times\cdots\times(2k-1)}$ A problem asks me to determine if the series
$$\sum_{k=1}^\infty \frac{2 \times 4 \times 6 \times \cdots \times (2k)}{1 \times 3 \times 5 \times \cdots \times (2k-1)}$$
converges or div... | $$\sum_{k=1}^\infty \frac{2 \times 4 \times 6 \times \cdots \times (2k)}{1 \times 3 \times 5 \times \cdots \times (2k-1)}=$$
$$\sum_{k=1}^\infty \frac{\prod_{j=1}^k(2j)}{\prod_{j=1}^k(2j-1)}=$$
$$\sum_{k=1}^\infty \prod_{j=1}^k\frac{(2j)}{(2j-1)}$$
The product $a_k = \prod_{j=1}^k\frac{(2j)}{(2j-1)}$ is
$$\left(1+\dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Characteristic equation of a recurrence relation? I am trying to find the general term of the following recurrence relation:
$$a_{n + 1} = \frac{1}{2}(a_{n} + \frac{1}{a_{n}})$$
where $a_1 = 3$. I'm failing to write the characteristic equation.
| Here is what I came upon:
If we have: $a_{n} = \frac{1}{2}(a_{n} + \frac{\lambda^2}{a_{n}})$, then it holds that: $\frac{a_{n} - \lambda}{a_{n} + \lambda} = (\frac{a_{1} - \lambda}{a_{1} + \lambda})^{2^{n-1}}$, solving for $a_{n}$ from the above example we have:
$$\frac{a_{n} - 2}{a_{n} + 2} = (\frac{3 - 2}{3 + 2})^{2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1928503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
Prove the inequality $\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{a^2+c^2}<1+\frac{\sqrt2}{2}$ Let $a,b,c -$ triangle side and $a+b+c=1$. Prove the inequality
$$\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{a^2+c^2}<1+\frac{\sqrt2}{2}$$
My work so far:
1) $a^2+b^2=c^2-2ab\cos \gamma \ge c^2-2ab$
2) $$\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{... | Try to use Lagrange multipliers to solve this problem.
max $[\sqrt{a^2+b^2}+\sqrt{c^2+b^2}+\sqrt{a^2+c^2}+\lambda(a+b+c-1)]$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1929245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Find the equation of two ...
Find the single equation of two straight lines that pass through the point $(2,3)$ and parallel to the line $x^2 - 6xy + 8y^2 = 0$.
My Attempt:
Let, $a_1x+b_1y=0$ and $a_2x+b_2y=0$ be the two lines represented by $x^2-6xy+8y^2=0$.
then,
$$(a_1x+b_1y)(a_2x+b_2y)=0$$
$$(a_1a_2)x^2+(a_1b_2+b... | The lines $$(x-2)^2 - 6(x-2)(y-3) + 8(y-3)^2 = 0$$ on transfer of origin to $(2,3)$, using the transformations $X = x-2, Y= y-2$ becomes $$X^2 - 6XY + 8Y^2 = 0$$ Hence the required equation is $$(x-2)^2 - 6(x-2)(y-3) + 8(y-3)^2 = 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solve $\iint_D\sqrt{9-x^2-y^2}$ Where $D$ is the positive side of a circle of radius 3
Solve $\displaystyle\iint_D\sqrt{9-x^2-y^2}$ Where $D$ is the positive
side of a circle of radius 3 ($x^2+y^2=9,x\ge0,y\ge0$)
I tried to subsitute variables to $r$ & $\theta$:
$$x = r\cos\theta$$
$$y = r\sin\theta$$
$$E = \{0\le ... | It just is the half of the capacity of the ball: ($x^2+y^2+z^2=9)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1930244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Finding the gradient of a given function Given the function $\phi=B_0/r=B_0/\vert \mathbf{x}\vert$ in a spherical axisymmetric geometry, where $B_0$ is a constant. Find $\nabla\phi$.
The given answer is
$$\phi=\frac{B_0}{r}=\frac{B_0}{\vert \mathbf{x}\vert}\implies \nabla\phi = -\frac{B_0\mathbf{x}}{\vert \mathbf{x}\... | 1º Note that r=$\sqrt{x^2+y^2+z^2}$, then ∇ϕ=($\frac{∂}{∂x}\frac{B_0}{\sqrt{x^2+y^2+z^2}}$,$\frac{∂}{∂x}\frac{B_0}{\sqrt{x^2+y^2+z^2}}$,$\frac{∂}{∂x}\frac{B_0}{\sqrt{x^2+y^2+z^2}}$).
2º$\frac{∂}{∂x}\frac{B_0}{\sqrt{x^2+y^2+z^2}}$=$-\frac{B_0}{2(\sqrt{x^2+y^2+z^2})^3} 2x$.
3º ∇ϕ=($-\frac{B_0}{\sqrt{x^2+y^2+z^2})^3} x,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1931395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Prove $\lim_{(x,y)\to(1,1)} x^2+xy+y=3$
Prove that $$\lim_{(x,y)\to(1,1)} x^2 + xy + y = 3$$ using the epsilon-delta definition.
What I have tried:
Let $\epsilon > 0$ be arbitrary. We must show that for every $\epsilon$ we can find $\delta>0$ such that
$$0 < \|(x,y) - (1,1)\| < \delta \implies \|f(x,y) - 3\| < \epsi... | \begin{align}
x^2+xy+y-3 &= (x-1)^2+2x-1+(x-1)(y-1)+x+y-1+(y-1)+1-3 \\
&=(x-1)^2+(x-1)(y-1)+(y-1)+3x+y-4 \\
&=(x-1)^2+(x-1)(y-1)+(y-1)+3(x-1)+(y-1)\\
&=(x-1)^2+(x-1)(y-1)+2(y-1)+3(x-1)\\
\end{align}
Let $\delta= \min(1, \frac{\epsilon}7),$
Then
\begin{align}
|x^2+xy+y-3| &\leq |x-1|^2+|x-1||y-1|+2|y-1|+3|x-1| \\
&\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1932891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Easier way to calculate the derivative of $\ln(\frac{x}{\sqrt{x^2+1}})$? For the function $f$ given by
$$
\large \mathbb{R^+} \to \mathbb{R} \quad x \mapsto \ln \left (\frac{x}{\sqrt{x^2+1}} \right)
$$
I had to find $f'$ and $f''$.
Below, I have calculated them.
But, isn't there a better and more convenient way to do t... | By implicit differentiation:
Let
$$
y(x) = \log\left[\frac{x}{\sqrt{x^2 + 1}}\right].
$$
Then
$$
(x^2 + 1)e^{2 y(x)} = x^2.
$$
Differentiating both sides,
$$
(x^2 + 1)e^{2y(x)}y'(x) + x e^{2y(x)} = x.
$$
Solving for $y'(x)$,
$$
y'(x) = \frac{x(e^{-2y(x)}-1)}{(x^2 + 1)} = \frac{1}{x(x^2+1)}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1933460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 5
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Prove that $a/(p-a) + b/(p-b) + c/(p-c) \ge 6$ Prove that $a/(p-a) + b/(p-b) + c/(p-c) \ge 6$ , where $a,b,c$ are the sides of a triangle and $p$ is the semi-perimeter .
| By C-S $\sum\limits_{cyc}\frac{a}{p-a}\geq\frac{(a+b+c)^2}{\sum\limits_{cyc}(pa-a^2)}=\frac{2(a+b+c)^2}{(a+b+c)^2-2(a^2+b^2+c^2)}\geq\frac{2(a+b+c)^2}{(a+b+c)^2-\frac{2}{3}(a+b+c)^2}=6$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Solving system of $9$ linear equations in $9$ variables I have a system of $9$ linear equations in $9$ variables:
\begin{array}{rl}
-c_{1}x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} &= 0 \\
x_{1} - c_{2}x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} &= 0 \\
x_{1} + x_{2} - c_{3}... | Let
$$\mathrm A := 1_n 1_n^T - \mbox{diag} (1 + c_1, \dots, 1 + c_n)$$
where $c_i \neq -1$ for all $i \in \{1,2,\dots,n\}$. Using the matrix determinant lemma,
$$\det (\mathrm A) = \left( 1 - \sum_{i=1}^n \frac{1}{1 + c_i} \right) (-1)^n \left( \prod_{i=1}^n (1+c_i)\right)$$
We want the homogeneous linear system $\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1934857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Determinant involving function of $x$ If $f(x)$ is a polynomial satisfying $$f(x)=\frac{1}{2}
\begin{vmatrix} f(x) & f(\frac{1}{x})-f(x) \\ 1 & f(\frac{1}{x}) \end{vmatrix} $$ and $f(3)=244$ then $f(2)$ is what?
My attempt—
Replacing $x$ by $\frac{1}{x}$ we get $$f\left(\frac{1}{x}\right)=\frac12\begin{vmatrix} f\left(... | Your starting idea is indeed great. Using $f(x)$ and $f(1/x)$ both, and adding them up, we have
$$
f(x)+f(1/x) = f(x)f(1/x).
$$
This gives
$$
f(1/x)-1=\frac{1}{f(x)-1}. \ \ \ \ (*)
$$
As $x\rightarrow\infty$, we have $f(0)=1$.
By assuming that $n=\textrm{deg}(f)$, we have by multiplying $x^n$ both sides,
$$
\frac{x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Proof that $ \|x+y\|^2 - \|x\|^2 \geq b(1 - 2^{-n})\|y\|^2 + 2^n( \|x+2^{-n}y\|^2 - \|x\|^2), \quad \forall x,y \in E, \forall n \in \mathbb N $ Suppose that $E$ is a Banach space over $\mathbb R$ satisfying the following inequality, for some $b > 0$
$$ \|x+y\|^2 + b\|x-y\|^2 \leq 2\|x\|^2 + 2 \|y\|^2, \quad \forall x,... | I think I might have solved the problem. Let's re-write (I) as
$$ \|x+y\|^2 - \|x\|^2 \geq b(\frac{2^n - 1}{2^n})\|y\|^2 + 2^n\|x+\frac{1}{2^n}y\|^2 - 2^n\|x\|^2 $$
$$ 2^n\|x+y\|^2 - 2^n\|x\|^2 \geq b(2^n - 1)\|y\|^2 + (2^n)^2\|x+\frac{1}{2^n}y\|^2 - (2^n)^2\|x\|^2 $$
$$ 2^n\|x+y\|^2 - 2^n\|x\|^2 \geq b(2^n - 1)\|y\|^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1935587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Explanations about some Mittag-leffler partial fraction expansions Is it possible to show where the following series come from?
$$\sum _{k=1}^{\infty } \left(\frac{1}{\pi ^2 k^2}-\frac{2}{(x-2 \pi k)^2}-\frac{2}{(2 \pi k+x)^2}\right)+\left(-\frac{2}{x^2}-\frac{1}{6}\right)=\frac{1}{\cos (z)-1}$$
$$\sum _{k=1}^{\infty... | As I commented earlier, I have a problem with the first expression. So, since Maple said that it is correct, I suppose I am wrong but I would like to know where.
Let me consider $$S_1=\sum _{k=1}^{\infty } \frac{1}{\pi ^2 k^2}\qquad S_2=\sum _{k=1}^{\infty }\frac{1}{(x-2 \pi k)^2}\qquad S_3=\sum _{k=1}^{\infty }\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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What is the value of the nested radical $\sqrt[3]{1+2\sqrt[3]{1+3\sqrt[3]{1+4\sqrt[3]{1+\dots}}}}$? The closed-forms of the first three are well-known,
$$x_1=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}\tag1$$
$$x_2=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\dots}}}}\tag2$$
$$x_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\dots... | To answer this question we need to find the system that is error resilient.
We can write the equation as
$$y(x)=x\sqrt[3]{1+(x+1)\sqrt[3]{1+(x+2)\sqrt[3]{1+...}}}$$
from where we have
$$y(x)=x\sqrt[3]{1+y(x+1)}$$
or
$$y(x-1)=(x-1)\sqrt[3]{1+y(x)}$$
Now we need to estimate how this function behaves and we can easily see... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1939394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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mimimum value of expression $a^2+b^2$
If $a,b$ are two non zero real numbers and $ab(a^2-b^2) = a^2+b^2,$ Then $\min(a^2+b^2)$
$\bf{My\; Try::}$ We can write it as $$ab=\frac{a^2+b^2}{a^2-b^2}\Rightarrow a^2b^2=\frac{(a^2+b^2)^2}{(a^2+b^2)^2-4a^2b^2}$$
Now Put $a^2+b^2=u$ and $a^2b^2=v,$ Then expression convert into ... | If we let $z = a + b i$ then we get
$$ab = \frac{ z^2 - \bar{z}^2}{4i}$$
$$a^2 - b^2 = \frac{ z^2 + \bar{z}^2}{2}$$
$$a^2 + b^2 = z \bar{z}$$
Thus your original equation can be rewritten as
$$\frac{Im(z^4)}{4} = \frac{z^4 - \bar{z}^4}{8 i} = \frac{(z^2 - \bar{z}^2)(z^2 + \bar{z}^2)}{8i} = z \bar{z}$$
We have $\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1941844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
$ \int\frac{2+\sqrt{x}}{\left(x+\sqrt{x}+1\right)^2}dx$ $$I=\int\frac{2+\sqrt{x}}{\left(x+\sqrt{x}+1\right)^2}dx$$
I can't think of a substitution to solve this problem, by parts won't work here. Can anyone tell how should I solve this problem?
| Let $$I = \int\frac{2+\sqrt{x}}{(x+\sqrt{x}+1)^2}dx = \int \frac{2+\sqrt{x}}{x^2\left(1+x^{-\frac{1}{2}}+x^{-1}\right)^2}dx$$
So $$I = \int\frac{2x^{-2}+x^{-\frac{3}{2}}}{\left(1+x^{-\frac{1}{2}}+x^{-1}\right)^2}dx$$
Put $\left(1+x^{-\frac{1}{2}}+x^{-1}\right) = t\;,$ Then $\displaystyle \left(-\frac{1}{2}x^{-\frac{3}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1942038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Use congruence's to find the reminder when $2^{50}$ and $41^{65}$ are divided by $7$ Use congruence's to find the reminder when $2^{50}$ and $41^{65}$ are divided by 7
$2^{50}$
$50=(7)^2+1$
$2^{50}=2^{7\cdot7+1}$
and I'm not sure where to go from here?
| Note that, $$\begin{align} & 2^3\equiv 1 \pmod7 \\ \implies & (2^3)^{16}\equiv 1^{16} \pmod7 \\ \implies & 2^{48}\equiv 1 \pmod7 \\ \implies & 2^{48}\cdot 2^2\equiv 1\cdot 2^2 \pmod7 \\ \implies & \color{blue}{2^{50}\equiv 4 \pmod7}\end{align}$$
Also note that $$\begin{align} & 41\equiv -1\pmod7 \\ \implies & 41^{65}\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1942297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that $(a\cos\alpha)^n + (b\sin\alpha)^n = p^n$ under the following conditions? How to prove that $(a\cos\alpha)^n + (b\sin\alpha)^n = p^n$ when then line $x\cos\alpha + y\sin\alpha = p$ touches the curve $$\left (\frac{x}{a} \right )^\frac{n}{n-1} + \left (\frac{y}{b} \right )^\frac{n}{n-1}=1$$
What I've ... | At the tangent point (x, y), the normal of the two curves are parallel, so we can get
$$\{\cos \alpha, \sin \alpha \}//\{\frac{x^\frac{1}{n-1}}{a^\frac{n}{n-1}}, \frac{x^\frac{1}{n-1}}{a^\frac{n}{n-1}}\} $$
so we get the equation:
$$\frac{x^\frac{1}{n-1}}{a^\frac{n}{n-1}\cos \alpha}=\frac{x^\frac{1}{n-1}}{a^\frac{n}{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1943891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How would you calculate the surface of the part of the paraboloid $z=x^2+y^2$ with $1 \le z \le 4$? Do you calculate it like done below? can you calculate it in another way?
$z=x^2+y^2$
Let
$x=\sqrt{z}\cos\theta$
$y=\sqrt{z}\sin\theta$
$z=z$
where $\theta\in[0,2\pi]$ and $z\in[1,4]$
$ \dfrac{\partial{x}}{\partial{\thet... | Your result is correct. By using polar coordinates, we obtain
$$\iint_S dS=\int_{\rho=1}^2 \int_0^{2\pi} \sqrt{1+f_x^2+f_y^2} \,(d\theta\,\rho d\rho)=2\pi\int_{\rho=1}^2 \sqrt{1+4\rho^2} \,\rho d\rho\\
=\dfrac{\pi}{6}\left[(1+4\rho^2)^{3/2}\right]_{\rho=1}^2
=\dfrac{\pi}{6}(17\sqrt{17}-5\sqrt{5})$$
where $f(x,y)=x^2+y... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find positive integer $x,y$ such that $7^{x}-3^{y}=4$ Find all positive integers $x,y$ such that $7^{x}-3^{y}=4$.
It is the problem I think it can be solve using theory of congruency. But I can't process somebody please help me .
Thank you
| Let us go down the rabbit hole. Assume that there is a solution with $ x, y > 1 $, and rearrange to find
$$ 7(7^{x-1} - 1) = 3(3^{y-1} - 1) $$
Note that $ 7^{x-1} - 1 $ is divisible by $ 3 $ exactly once (since $ x > 1 $): the contradiction will arise from this.
Reducing modulo $ 7 $ we find that $ 3^{y-1} \equiv 1 $, ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a series approximation in terms of $n$ for the sum of the harmonic progression : $\sum_{k=0}^{n}\frac{1}{1+ak}$? When $a=1$ the sum is given by $ H_{n} $ and we have : $$H_{n}=log(n)+γ+\frac{1}{2n}-\frac{1}{12n^2}+\frac{1}{120n^4} \hspace{0.5cm}.\hspace{.1cm}.\hspace{.1cm}. $$ Does any representation of a simi... | As Felix Marin answered,$$\sum_{k = 1}^{n}{1 \over 1 + ak}=\frac{H_{n+\frac{1}{a}}-H_{\frac{1}{a}}}{a}$$ Now, using the asymptotics $$H_m=\gamma +\log \left({m}\right)+\frac{1}{2 m}-\frac{1}{12
m^2}+\frac{1}{120 m^4}+O\left(\frac{1}{m^5}\right)$$ $$H_{n+\frac{1}{a}}=\gamma +\log \left({n+\frac{1}{a}}\right)+\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1948139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Find the values of $\alpha $ satisfying the equation(determinant)
Find the values of $\alpha $ satisfying the equation
$$\begin{vmatrix}
(1+\alpha)^2 & (1+2\alpha)^2 & (1+3\alpha)^2\\
(2+\alpha)^2& (2+2\alpha)^2 & (2+3\alpha)^2\\
(3+\alpha)^2& (3+2\alpha)^2 & (3+3\alpha)^2
\end{vmatrix}=-648\alpha $$
I used ... | Hint write it as a product of two determinants after taking $\alpha,\alpha^2$ common from one of the determinants to get $\alpha=\pm 9$ or to continue your method use $R_1\to R_1-R_2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1948216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does $\sum_{n=0}^k \cos^{2k}\left(x + \frac{n \pi}{k+1}\right) = \frac{(k+1)\cdot(2k)!}{2^{2k} \cdot k!^2}$? In the paper "A Parametric Texture Model based on Joint Statistics of Complex Wavelet Coefficients", the authors use this equation for the angular part of the filter in polar coordinates:
$$\sum_{n=0}^k \cos... | Suppose we seek to verify that
$$\sum_{k=0}^n \cos^{2n}\left(x+\frac{k\pi}{n+1}\right)
= \frac{n+1}{2^{2n}} {2n\choose n}.$$
The LHS is
$$\sum_{k=0}^n \cos^{2n}\left(x+\frac{k\times 2\pi}{2n+2}\right).$$
Observe also that
$$\sum_{k=0}^n
\cos^{2n}\left(x+\frac{(k+n+1)\times 2\pi}{2n+2}\right)
\\ = \sum_{k=0}^n
\cos^{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1951708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prime modulo maximum Prove that the remainder of division of a positive number $n$ by a prime $p \le n$ is maximized when $p$ is the smallest prime larger than $\frac{n}{2}.$
It is easy to see that for any number of the form $\frac{n}{2}+k$ where $k \gt 0$, if $k$ is increased remainder will decrease. How to prove th... | This does not hold true. Consider $n=14$, the smallest prime larger than $\frac{14}{2}=7$ is $11$, but the maximum remainder is attained for prime $5 \lt 7$:
$$14 \bmod 5 = 4 \;\;\gt\;\; 14 \bmod 11 = 3$$
[ EDIT ] The following shetches the proof to the related question asked in a comment below.
The remainder of ... | {
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"timestamp": "2023-03-29T00:00:00",
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show that $X/(X+Y) $ has a cauchy distribution if $X $ and $X+Y $ are standard normal A random variable $X$ has a cauchy distribution with parameters $a$ and $b$ is the density of $X$ is $f(x\mid a,b)=\dfrac{1}{\pi b}\dfrac{1}{1+(\frac{x-a}{b})^2}$ where $-\infty <x< \infty $, $-\infty <a <\infty$, $b>0$
Suppose $X$ a... | You had almost reached the end. It suffices to transform:
$$\left(\dfrac{1}{2\pi}\right)\dfrac{1}{v^2-v+\frac{1}{2}}=\left(\dfrac{1}{2\pi}\right)\dfrac{1}{(v-\frac{1}{2})^2+(\frac{1}{2})^2}=\left(\dfrac{1}{2\pi}\right)\dfrac{4}{1+\left(\frac{v-\frac{1}{2}}{\frac{1}{2}}\right)^2}$$
giving
$$\left(\dfrac{1}{\pi \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How can the surd $\sqrt{2-\sqrt{3}}$ be expressed? I was wondering how $\sqrt{2-\sqrt{3}}$ could be expressed in terms of $\frac{\sqrt{3}-1}{\sqrt{2}}$.
I did try to solve both the expressions separately but none of them seemed to match.
I would appreciate it if someone could also mention the procedure
| Theorem: Given a nested radical of the form $\sqrt{X\pm Y}$, it can be rewritten into the form $$\sqrt{\frac {X+\sqrt{X^2-Y^2}}{2}}\pm\sqrt{\frac {X-\sqrt{X^2-Y^2}}{2}}\tag{1}$$
Where $X>Y$.
Therefore, we have $X=2,Y=\sqrt{3}$ because $2>\sqrt{3}$. So plugging that into $(1)$ gives us $$\sqrt{\frac {2+\sqrt{4-3}}{2}}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1954822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Number of integer triangles with sides 4 or less
Consider triangles having integer sides such that no side is greater than 4 units. How many such triangles are possible?
I suspect a relation to the following question:
How many ways can $r$ things be taken from $n$ with repetition and without regard to order?
| Without loss of generality, assume that $a \leq b \leq c < a+b$. This leaves us with very few choices.
1) All three could be equal. That gives us four choices.
2) $a=1$. Then, $b+1 > c$, so $b=c$ must happen, this gives three choices.
3) $a=2$. Then, $b \leq c < b+2$, so $b=2$,$c=3$ and $b=3, c=3,4$ , and $b=c=4$ are t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1956257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $S = \{a^2 + b^2: a,b \in \Bbb N\}$ is closed under multiplication. Is it true?
Can you prove or disprove this?
$S = \{a^2 + b^2: a,b \in \Bbb N\}$ is closed under multiplication.
| Suppose that $x=a^2+b^2$ and $y=c^2+d^2$. Then
$$xy=\det
\begin{pmatrix}
a&b\\-b&a
\end{pmatrix}
\det\begin{pmatrix}
c&d\\-d&c
\end{pmatrix}
=\det
\begin{pmatrix}
a&b\\-b&a
\end{pmatrix}
\begin{pmatrix}
c&d\\-d&c
\end{pmatrix}
=\det\left(
\begin{array}{cc}
a c-b d & b c+a d \\
-b c-a d & a c-b d \\
\end{array}
\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1956926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Question in complex numbers from GRE This is a question motivated from GRE subgect test exam.
if f(x) over the real number has the complex numbers $2+i$ and $1-i$ as roots,then f(x) could be:
a) $x^4+6x^3+10$
b) $x^4+7x^2+10$
c) $x^3-x^2+4x+1$
d) $x^3+5x^2+4x+1$
e) $x^4-6x^3+15x^2-18x+10$
What I thought at first was ... | 1.
$(x-(2-i))(x-(2+i))$
$x^2-x(2+i)-x(2-i)+(2-i)(2+i)$
$x^2-2x-xi-2x+xi+(4-2i+2i+1)$
$x^2-4x+5$
2.
$(x-(1-i))(x-(1+i))$
$x^2-x(1+i)-x(1-i)+(1-i)(1+i)$
$x^2-x-xi-x+xi+(1+i-i+1)$
$x^2-2x+2$
3.
$(x^2-4x+5)(x^2-2x+2)$
$x^4-2x^3+2x^2-4x^3+8x^2-8x+5x^2-10x+10$
$x^4-6x^3+15x^2-18x+10$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1958030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Prove that $(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$
Let $a,b,c,d>0$. Prove that $$(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$$
I don't know how to begin to solve this problem
| We can assume that $a+b+c+d=2$. Then the inequality becomes
$$
(1-d)(1-c)(1-b)(1-a)\le\left(1-\tfrac{c+d}2\right)\left(1-\tfrac{a+d}2\right)\left(1-\tfrac{a+b}2\right)\left(1-\tfrac{c+b}2\right)\tag{1}
$$
If any of $a$, $b$, $c$, or $d$ is greater than $1$, then the left side is negative and the inequality is trivial. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1959149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Calculating $\sum_{n=1}^∞ \frac{1}{(2 n-1)^2+(2 n+1)^2}$ using fourier series of $\sin x$ I have to calculate $\frac{1}{1^2+3^2}+\frac{1}{3^2+5^2}+\frac{1}{5^2+7^2}+...$ using half range Fourier series $f(x)=\sin x$ which is:
$f(x)=\frac{2}{\pi}-\frac{2}{\pi}\sum_{n=2}^\infty{\frac{1+\cos
n\pi}{n^2-1}\cos nx}$
I hav... | A different approach. Since $\frac{1}{(2n-1)^2+(2n+1)^2}=\frac{1}{8n^2+2}$ we have:
$$\begin{eqnarray*} \sum_{n\geq 1}\frac{1}{(2n-1)^2+(2n+1)^2}&=&\frac{1}{8}\sum_{n\geq 1}\frac{1}{n^2+\frac{1}{4}}\\&=&\frac{1}{8}\int_{0}^{+\infty}\sum_{n\geq 1}\frac{\sin(nx)}{n}e^{-x/2}\,dx\\&=&\frac{1}{8(1-e^{-\pi})}\int_{0}^{2\pi}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Verify the triple angle formula Verify the triple angle formula
$$\tan(3x) = \frac{3 \tan(x) − \tan^3(x)}{1 − 3 \tan^2(x)}$$
I have tried simplifying the right side by the following
$$\tan(3x) = \frac{\tan(x)(3 − \tan^2(x))}{1 − 3 \tan^2(x)}$$
but then I am getting stuck trying to verify the equation
| $$\tan 3x =\tan(x+2x)= \frac{\tan x + \tan 2x}{1 − \tan x \tan 2x}=$$
$$=\frac{\tan x + \frac{2\tan x}{1-\tan^2x}}{1 − \tan x \frac{2\tan x}{1-\tan^2x}}=\frac{3\tan x- \tan ^3x}{1-3\tan^2x}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Limit of the two variable function $f(x,y).$ How to show that $$\lim_{(x,y)\to(0,0)}\frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}(x^{4}+y^{2})}=0$$ I tried with different paths as $x=0,y=0, y=x$ its comes to zero but i have no general idea. Please help. Thanks to lot.
| You have
\begin{align}
\left| \frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}(x^{4}+y^{2})}\right|
&=\frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}(x^{4}+y^{2})}
\leq
\frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}\,(y^{2})}\\ \ \\
&=\frac{x^2}{\sqrt{x^2+y^2}}\leq\frac{x^2}{\sqrt{x^2}}=|x|\\ \ \\
&\leq\sqrt{x^2+y^2}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1963119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does the sum $\sum_{n \geq 1} \frac{2^n\operatorname{mod} n}{n^2}$ converge? I am somewhat a noob, and I don't recall my math preparation from college. I know that the sum $\displaystyle \sum_{n\geq 1}\frac{1}{n}$ is divergent and my question is if the sum$$\sum \limits _{n\geq 1}\frac{2^n\mod n}{n^2}$$converges. I thi... | In this answer, we prove that
$$ \sum_{n=1}^{\infty} \frac{2^n \text{ mod } n}{n^2} = \infty. \tag{*} $$
Idea. The intuition on $\text{(*)}$ comes from the belief that the sequence $(2^n \text{ mod } n)/n$ is equidistributed on $[0, 1]$, which is quite well supported by numerical computation.
Proving this seems quite ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
"answer_count": 3,
"answer_id": 1
} |
$4x^2y′′-8x^2y′+(4x^2+1)y=0$ solve by Frobenius Method. I would like to ask if someone can explain to me how can we solve the following DE using this method.
$4x^2y′′-8x^2y′+(4x^2+1)y=0$
| Given differential equation $4x^2y^{\prime\prime}-8x^2y^{\prime}+(4x^2+1)y=0$. $P(x)=-2$
and $Q(x)=\frac{1+4x^2}{4x^2}$, since $Q(x)$ is not analytic at $x=0$, we say $x=0$ is a
singular point of this differential equation . Singular points are futher classified into regular singular
and irregular singular points. S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is $\lim\limits_{n\to\infty}\frac {1}{n^2}\sum\limits_{k=0}^{n}\ln\binom{n}{k} $? It was originally asked on another website but nobody has been able to prove the numerical result. The attempts usually go by Stirling's approximation or try to use the Silverman-Toeplitz theorem.
| By Stolz Cezaro
$$\lim\limits_{n\to\infty}\frac {1}{n^2}\sum\limits_{k=0}^{n}\ln\binom{n}{k}=\lim\limits_{n\to\infty}\frac {1}{2n+1} \left(\sum\limits_{k=0}^{n+1}\ln\binom{n+1}{k}- \sum\limits_{k=0}^{n}\ln\binom{n}{k} \right)\\
=\lim\limits_{n\to\infty}\frac {1}{2n+1} \sum\limits_{k=0}^{n}\left(\ln\binom{n+1}{k}- \ln\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1966057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Diophantine equation $\frac{a+b}{c}+\frac{b+c}{a}+\frac{a+c}{b} = n$ Let $a,b,c$ and $n$ be natural numbers and $\gcd(a,b,c)=\gcd(\gcd(a,b),c)=1$.
Does it possible to find all tuples $(a,b,c,n)$ such that:
$$\frac{a+b}{c}+\frac{b+c}{a}+\frac{a+c}{b} = n?$$
| First note that we can assume $a,b,c \in \mathbb{Q}$, without loss of generality. A rational solution can then be scaled to integers.
Combining into a single fraction gives the quadratic in $a$
\begin{equation*}
a^2+\frac{b^2-nbc+c^2}{b+c}a+bc=0
\end{equation*}
and, for rational solutions, the discriminant must be a ra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Probability that a number is divisible by 11 The digits $1, 2, \cdots, 9$ are written in random order to form a nine digit number. Then, the probability that the number is divisible by $11$ is $\ldots$
I know the condition for divisibility by $11$ but I couldn't guess how to apply it here.
Please help me in this regard... | The rule of divisibility by $11$ is as follows:
The difference between the sum of digits at odd places and the sum of the digits at even places should be $0$ or a multiple of $11$.
We also know that the sum of all the digits will be $45$ as $1 + 2 + ... + 9 = 45$.
Let $x$ denote sum of digits at even position s and $y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 2
} |
A better way to evaluate a certain determinant
Question Statement:-
Evaluate the determinant:
$$\begin{vmatrix}
1^2 & 2^2 & 3^2 \\
2^2 & 3^2 & 4^2 \\
3^2 & 4^2 & 5^2 \\
\end{vmatrix}$$
My Solution:-
$$
\begin{align}
\begin{vmatrix}
1^2 & 2^2 & 3^2 \\
2^2 & 3^2 & 4^2 \\
3^2 & 4^2 & 5^2 \\
\end{vmatrix} &=
(1^2\t... | Using the rule of Sarrus, the computation is really not too long, and we get in general for all $n\ge 1$,
$$
\det \begin{pmatrix} n^2 & (n+1)^2 & (n+2)^2\cr (n+1)^2& (n+2)^2 & (n+3)^2\cr
(n+2)^2& (n+3)^2 & (n+4)^2\end{pmatrix}=-8.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1969290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 4
} |
How to evaluate $\lim\limits_{x\to 0} \frac{\arctan x - \arcsin x}{\tan x - \sin x}$ I have a stuck on the problem of L'Hospital's Rule,
$\lim\limits_{x\to 0} \frac{\arctan x - \arcsin x}{\tan x - \sin x}$ which is in I.F. $\frac{0}{0}$
If we use the rule, we will have
$\lim\limits_{x\to 0} \frac{\frac{1}{1+x^2}-\fra... | Alternatively, one may use standard Taylor expansions, as $x \to 0$,
$$
\begin{align}
\sin x&=x-\frac{x^3}{6}+o(x^4)
\\\tan x&=x+\frac{x^3}{3}+o(x^4)
\\\arctan x&=x-\frac{x^3}{3}+o(x^4)
\\\arcsin x&=x+\frac{x^3}{6}+o(x^4)
\end{align}
$$ giving, as $x \to 0$,
$$ \frac{\arctan x - \arcsin x}{\tan x - \sin x}= \frac{-\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
System of quadratic equations with parameter I'd appreciate your help with this problem:
$p - a^2 = b$
$p - b^2 = c$
$p - c^2 = d$
$p - d^2 = a$
Where $a$, $b$, $c$, $d$ are real numbers, $p$ is a real parameter lower or equal to 1 and greater or equal to 0.
Thank you a lot. I am capable of solving the problem for pos... | *
*Subtracting two adjacent equations,
$$
\left \{
\begin{array}{ccc}
(a-b)(a+b) &=& c-b \\
(b-c)(b+c) &=& d-c \\
(c-d)(c+d) &=& a-d \\
(d-a)(d+a) &=& b-a
\end{array}
\right.$$
Note that
$$a=b \iff b=c \iff c=d \iff a=d$$
Now
$$a^2+a-p=0$$
$$a=b=c=d=\frac{-1\pm \sqrt{1+4p}}{2}$$
*
*Subtrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Show that $f(g(x)) = x$ and $g(f(x))= x$, with $f(x) = x^e \bmod n$ and $g(x) = x^d \bmod n$ I want to solve the following problem:
Let $d$ and $e$, both natural numbers, be each others inverses modulo $\varphi(n)$, where $n = p\cdot q$ is a product of two different prime numbers $p$ and $q$. Let $M = \{0,1,2,\dots,(n... | I think I figured it out. First I have to prove that $x^{k\varphi(n) + 1} \equiv x \pmod{n}$, even when I don't know if $\gcd(x,n) = 1$.
We look at the system
\begin{align}
\begin{cases}
y \equiv x \pmod p \\
y \equiv x \pmod q
\end{cases}
\end{align}
Since $q$ and $p$ are two different prime numbers, they are rel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1977611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
trouble with double/iterated integration of $\int^1_0[\int^1_0v(u+v^2)^4du]dv$ I have:
For $\int^1_0v(u+v^2)^4du$:
u substitution (using x instead since there's a u in there already) with $x=(u+v^2), dx/du=1$
$v\int^1_0x^4=v\frac{1}{5}x^5=v\frac{(u+v^2)^5}{5}-\frac{v(0+v^2)^5}{5}|^1_0=\frac{(v+v^3)^5}{5}-\frac{v^{15}}{... | An alternate approach would be to change the order of integration. This will greatly reduce the difficulty of the integral. As the limits of the inner integral do not depend upon $v$ this trivially becomes:
$$\int^1_0[\int^1_0v(u+v^2)^4\ dv]\ du$$
$$=\int^1_0\left[\frac{1}{10}\bigg((u+v^2)^5\bigg)_0^1\right]\ du$$
$$=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1979273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = \frac{1+\sqrt{5}}{2} = \phi$, is this a coincidence? I was playing around with square roots today when I "discovered" this.
$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = x$
$\sqrt{1 + x} = x$
$1 + x = x^2$
Which, via the quadratic formula, leads me to the golden ratio.
Is there an... | If you play around a little more, you will also notice that:
$$
\frac{1+\sqrt{5}}{2} = 1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+ \ldots} } } }
$$
Which simplifies to $x = 1+ \frac 1x \implies x^2=x+1$.
It's no coincidence. I mean to say, it comes directly from the equation itself.
Just to give you another ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1980909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Are these derivatives correct (the respective functions involve a square root, fraction and expansion)? Question: Use rules of differentiation to answer the following. There is no need to simplify your answer.
*
*If $y = 3x{\sqrt x}$, find $\frac{dy}{dx}$
My working:
$y = 3x^{1+\frac{1}{2}}$
$y = 3x^{\frac{3}{2}}... | Your work for Question 1 is fine. For Question 2, use the Quotient Rule (you seem to have assumed that the derivative of a fraction is the derivative of the numerator over the derivative of the denominator, which isn't the case):
\begin{align}
f' \left ( x \right ) & = \frac{5x \cdot 2 - \left ( 2x - 1 \right ) \cdot 5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the number that, when divided by $3$, $5$, $7$, leaves remainders of $2$, $3$, $2$, respectively? What is the number?
The LCM of the divisors is 105. I think this has something to do with the Chinese remainer theorem, but I am not sure how to apply this knowledge.
| We can solve in a simple way without the Chinese Remainder Theorem.
Let $n = 3x + 2 = 5y + 3 = 7z + 2$.
Then, $5y + 3 \equiv 3x + 2 \equiv (\mod 3) \implies y \equiv 1(\mod 3)y$
Hence let $y = 3k + 1$ which gives $n = 5y + 3 = 15k + 8$.
Again $n = 15k + 8 \equiv 7z + 2 (\mod 7) \implies k \equiv 1 (\mod 7)$
Hence let ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1984457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Calculate the sum of the series $\sum_{n=1}^\infty \frac 1{n(n+1)^2}$ I have already proven that
$$\int_0^1 \ln(1-x)\ln (x)\mathrm d x=\sum_{n=1}^\infty \frac 1{n(n+1)^2}.$$
So now I want so calculate
$$S:=\sum_{n=1}^\infty \frac 1{n(n+1)^2}.$$
I think I should use the fact that
$$\sum_{n=1}^\infty \frac 1{n^2}=\fra... | we have that
$$\sum_{n=1}^{m }\frac{1}{n}- \frac{1}{n+1}=\frac{m}{m+1}=1-\frac{1}{m+1}$$ the sum is $1$when the $m\rightarrow \infty$
$$\sum_{n=1}^{\infty }\frac{1}{n(n+1)^2} = \sum_{n=1}^{\infty }(\frac{1}{n}- \frac{1}{n+1}) -\sum_{n=1}^{\infty } \frac{1}{(n+1)^2}=1 -\sum_{n=2}^{\infty } \frac{1}{n^2}$$
$$=1-(\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1986102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$x + y = 7 $and $x^3 - y^3 = 37$, find $xy$
$x+y=7$
$x^3-y^3 = 37$
Find $xy$
I have tackled this one but I am stuck, when I get to this point:
$$(x-y)((x+2)^2 -xy) = 37$$
I would be glad if you could give me any suggestions.
| Since $(x^3-y^3)^2$ is symmetric in $x$ and $y$, you can express it as a polynomial in terms of $x+y$ and $xy$ :
$(x^3-y^3)^2 = x^6 - 2x^3y^3 + y^6 = (x+y)^6 - (6x^5y+15x^4y^2+22x^3y^3+15x^2y^4+6xy^5) \\
= (x+y)^6 - 6xy(x+y)^4 + (9x^4y^2+14x^3y^3+9x^2y^4) \\
= (x+y)^6 - 6xy(x+y)^4 + 9(xy)^2(x+y)^2 - 4(xy)^3$
Plugging t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1986213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $a^2$ when divided by $5$ cannot have a remainder of $3$. For any natural number $a$, prove that $a^2$ when divided by $5$ cannot have a remainder of $3$. (Hint: What are the possible values of the remainder when $a$ is divided by $5$?
Using the hint, I found that the possible values of the remainder are $1,... | if $a \equiv 0 \mod 5$, then $a^2 \equiv 0 \mod 5$.
if $a \equiv 1 \mod 5$, then $a^2 \equiv 1 \mod 5$.
if $a \equiv 2 \mod 5$, then $a^2 \equiv 4 \mod 5$.
if $a \equiv -2 \mod 5$, then $a^2 \equiv 4 \mod 5$.
if $a \equiv -1 \mod 5$, then $a^2 \equiv 1 \mod 5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Prove: $ \frac{1}{\sin 2x} + \frac{1}{\sin 4x } + \cdots + \frac{1 }{\sin 2^n x} = \cot x - \cot 2^n x $
Prove
$$ \frac{1}{\sin 2x} + \frac{1}{\sin 4x } + \cdots + \frac{1 }{\sin 2^n x} = \cot x - \cot 2^n x $$
where $n \in \mathbb{N}$ and $x$ not a multiple of $\frac{ \pi }{2^k} $ for any $k \in \mathbb{N}$.
My... | $$\csc2A+\cot2A=\dfrac{1+\cos2A}{\sin2A}=\cot A$$
$$\iff\csc2A=\cot A-\cot 2A$$
Put $2A=2x,4x,8x,\cdots,2^nx$ and add
See also: Telescoping series
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1987415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Let f be an arbitrary, twice differentiable function for which $f''\neq0$ Let f be an arbitrary, twice differentiable function for which $f''\neq0$ . The function $u(x,y)=f(x^2+axy+y^2)$ satifsfies equation $U_{xx}-U_{yy}=0$ then the constant a is.
$f_{xx}(x^2+axy+y^2)(2x+ay)^2+2f_x(x^2+axy+y^2)-f_{yy}(x^2+axy+y^2)(2y... | By the chain rule, $$U_{xx} = 2f'(x^2+axy+y^2) + (2x+ay)^2f''(x^2+axy+y^2)$$ and $$U_{yy} = 2f'(x^2+axy+y^2) + (2y+ax)^2f''(x^2+axy+y^2).$$
Since $U_{xx}-U{yy}=0$, and $f''(x) \neq 0, \forall x \in D_f$, we have $$((2x+ay)^2-(2y+ax)^2)f''(x^2+axy+y^2) = 0 \Rightarrow 4x^2+4axy+a^2y^2 - 4y^2 -4axy - a^2x^2 = 0.$$
In par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1989883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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real values of $a$ for which the range of function $ f(x) = \frac{x-1}{1-x^2-a}$ does not contain value from $\left[-1,1\right]$
All real values of $a$ for which the range of function $\displaystyle f(x) = \frac{x-1}{1-x^2-a}$ does not
contain any value from $\left[-1,1\right]$
$\bf{My\; Try::}$ Let $\displaystyle y ... | Given
$$f(x)=\dfrac{x-1}{1-x^2-a}$$
for some $a$
notice that if for $x\ne0$ you multiply both the numerator and denominator by $\dfrac{1}{x}$ you obtain
$$ f(x)=\dfrac{1-\frac{1}{x}}{\frac{1}{x}-x-\frac{a}{x}}\text{ for }x\ne0 $$
Now notice what happens to the value of $f(x)$ for large values of $x$. Regardless of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1991546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How many #8 digits can you build?
How many different 8-digit numbers can be formed using two 1s, two 2s, two 3s, and two 4s such that no two
adjacent digits are the same?
So we have $0 -> 9 = 10$ options for the digits but with the constraints we have something different.
Cases:
$1$ _ _
$2$ _ _
$3$ _ _
$4$ _ _
W... | Use Inclusion-exclusion principle.
There are $\frac{8!}{2^4}$ 8-digits numbers that can be formed using two 1s, two 2s, two 3s, and two 4s.
i) How many of these 8-digits numbers have two adjacent 1s?
There are $7\cdot \frac{6!}{2^3}$ such numbers.
ii) How many of these 8-digits numbers have two adjacent 1s and two ad... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1992775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proof that $\prod_{k=1}^{n-1}(1+\frac1k)^k = \frac{n^n}{n!}$ for all $n \in \Bbb N \ge 2$ I've tried to prove this for a while now, but I can't get it:
$\prod_{k=1}^{n-1}(1+\frac1k)^k = \frac{n^n}{n!}$ for all $n \in \Bbb N \ge 2$
Solution:
$\prod_{k=1}^{(n+1)-1}(1+\frac1k)^k=\frac{(n+1)^{(n+1)}}{(n+1)!}$
$\left(1+\fra... | The following telescopic product in disguise:
$$ \prod_{k=1}^{n-1}\left(1+\frac{1}{k}\right) = n \tag{1}$$
leads to
$$ \prod_{k=1}^{n-1}\left(1+\frac{1}{k}\right)^k = \frac{n^n}{\prod_{k=1}^{n-1}\left(1+\frac{1}{k}\right)^{n-k}}=\frac{n^n}{n\cdot(n-1)\cdot\ldots\cdot 1}=\frac{n^n}{n!}.\tag{2}$$
In the opposite directio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1996735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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What is $(-1)^{\frac{1}{7}} + (-1)^{\frac{3}{7}} + (-1)^{\frac{5}{7}} + (-1)^{\frac{9}{7}} + (-1)^{\frac{11}{7}}+ (-1)^{\frac{13}{7}}$? The question is as given in the title. According to WolframAlpha, the answer is 1, but I am curious as to how one gets that. I tried simplifying the above into $$6(-1)^{\frac{1}{7}}$$ ... | We have
$$x^7+1=(x+1)(x^6-x^5+x^4-x^3+x^2-x+1).$$ Let $x=(-1)^{1/7}$ and get the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2003317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Parametrization of the intersection between a sphere and a plane I can't find a way to get the parametric equation $\gamma(t)=(x(t),y(t),z(t))$ of a curve that is the intersection of a sphere and a plane (not parallel to any coordinate planes). That is
$$\begin{cases} x^2+y^2+z^2=r^2 \\ ax+by+cz=d \end{cases}$$
I don'... | Just to present a vectorial approach to the problem
Consider the plane equation written as:
$$
\frac{{a\,x + b\,y + c\,z}}{{\sqrt {a^{\,2} + b^{\,2} + c^{\,2} } }} = \frac{d}{{\sqrt {a^{\,2} + b^{\,2} + c^{\,2} } }}
$$
That means that the points ${\bf p} = \left( {x,y,z} \right)$ on the plane
shall project onto th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2004224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.