# Tagged Questions

A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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### Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$

$\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$ is approximately $.5229461921333351$ but I've been assured that there is a closed form for this ...
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### Formula for number of faces in 4 dimensions

If a polytope has $m$ faces in 3 dimensions, how many faces does its analogous polytope have in four dimensions? Does a formula exist? For example, if $m=4$, you have a tetrahedron, and the 4-...
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### Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$

Does a closed form exist for $$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$ in terms of $k$ and other functions? The best that I have been able to do is solve the case where $k=1$, since the ...
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### Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured

By trial and error I have found numerically $$\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}=\frac{1}{6}-\frac{1}{2\pi}$$ how can this result be derived analytically?
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### Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
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### Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma$

Let $\gamma$ be the Euler-Mascheroni constant. I'm trying to prove that $$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma$$ I tried introducing a parameter to the ...
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### Solve integral $\int_{-1}^{1} \frac{dx}{(e^x+1)(x^2+1)}$

Solve following integral: $$\int_{-1}^{1} \frac{dx}{(e^x+1)(x^2+1)}$$ I tried various methods but without success.
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### A question on the Laplace Transform of $f(t)=t e^{at}\sin (bt)$ [closed]

I would like to solve the Laplace transform of the following function: $$t \mapsto t e^{at}\sin (bt).$$ I know that $\mathscr{L}\left(\sin(bt)\right)=\dfrac{b}{s^2+b^2}$ and that you have to ...
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### $\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$, for complex variable $z$.

I want to find this limit for complex variable $z$ $$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$$ In the real case I know $\sin(z)$ is bounded by $-1, 1,$ and the limit is $0$. But in the complex case ...
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### How to prove $\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right)$?

I need help, on how to prove $$\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right).$$ Any hints?
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### Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1}$

As the question says, $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1}$$ where a is a constant, $a>0$.
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### Double sum with binomial coefficients $\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$

Find a closed form formula for this sum: $$\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$$ It's quite likely that it can be done ...
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### Summation of $\sum_{n=0}^\infty e^{-\sqrt n}$

Is there a closed form for the following sum? $$\sum_{n=0}^\infty e^{-\sqrt n}$$
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### Help with $\int_0^\infty x^me^{-ax^n}dx$

I need the solution of the following integral $$\int_0^\infty x^me^{-ax^n}dx$$ where $a,n,m$ are all positive constants with $n\geq 2$. I have searched for it in the Gradshteyn but was unable to find ...
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### What is the method for solving this recurrence relation?

I have an equation for generating square-triangle numbers using a recurrence relation: $$f(n)^2+f(n)(2-34f(n-1))+(f(n-1)^2-70f(n-1)+1) = 0$$ But I wish to solve the equation to produce a closed form ...
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### How can I solve this recurrence relation for generating triangle-squares?

$$N_k = 17N_{k-1} + 6(8N^2_{k-1} + N_{k-1})^{1/2} + 1$$ $$k\geqslant 1$$ I'm trying to convert a recurrence relationship for producing triangle square-numbers into a closed-form expression in terms of ...
### Closed form of function $f(n) = (1/n) \sum _{x=1}^{n-1} f(x)$ [closed]
Could anyone help me get to the closed form of the function: $$f(n) = \frac 1 n \sum _{x = 1}^{n-1}f(x)$$ $$f(1) = 1$$