# 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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### Fourier coefficient and computing an improper integral

I am having difficulties with this problem. I don't really know where to start, I suspect there is something I am supposed to know or "see" that I am missing. $u$ is a $2\pi$-periodic function ...
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### Generating function for $\sum_{n=0}^{\infty} n^k x^n$

I would like to get the closed form for this generating function, assuming the $k$ is given up front / held constant: $\sum_{n=0}^{\infty} n^k x^n$ However I don't know if this is too advanced for me ...
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### How to evaluate $\int\frac{dx}{(2\sin x+\sec x)^4}$?

I tried a lot but I am not able to get a start. Can anyone give me the start of this question $$\int\frac{dx}{(2\sin x+\sec x)^4} \ ?$$
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### Solving an equation involving an integral: $\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$

Determine a pair of number $a$ and $b$ for which $$\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$$ I tried putting $x$ as $1-x$ as the integral wouldn't change but could not move forward from ...
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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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### 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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### Is there a closed form for this binomial sum?

I am looking for a closed form of this sum:$\sum\limits_{j=k}^n\binom{j}{k}(-1)^j$ I know that this sum has a closed form: $\sum\limits_{j=k}^n\binom{j}{k}=\binom{n+1}{k+1}$ I can get this closed ...
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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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### Closed form of $\int_0^1 \tan(\gamma\sqrt{1-x^2}) dx$

Some context: I'm studying the problem of nonperturbative pair creation from strong fields in quantum electrodynamics. For certain time dependent electric fields I can get some information about the ...
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### Problem concerning the sequence $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$

The question is : Prove that the sequence $\{s_n\}$ where $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$ is convergent. Hence find $\lim_{n \to \infty} \left(1 - 1/2 + 1/3 - ... - 1/2n\right)$. I have ...
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### A limit using the Euler number: $\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$

What is answer of this limit and how can I get it? $c$ and $i$ are constants. $$\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$$ I guess it will envolve some Neper/the Euler ...
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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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### 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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### A closed form of the series $\sum_{n=1}^{\infty} q^n \sin(n\alpha)$

I am having problems with the following series: $$\sum_{n=1}^{\infty} q^n \sin(n\alpha), \quad|q| < 1.$$ No restrictions on $\alpha$. I need to find out whether it converges and if yes, ...
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### 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: $$f(n) = \frac 1 n \sum _{x = 1}^{n-1}f(x)$$ $$f(1) = 1$$
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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 ...
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### Evaluating $\int_1^{\infty}x\: \text{erfc}(a+b \log (x)) \, dx$

I am trying to evaluate the following integral $$I = \int_1^{\infty } x \mathop{erfc}(a + b \log (x)) \, dx$$ where $a$, $b$ are some positive constants. Using the substitution $t = \log (x)$, ...
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### Does there exist a closed-form expression for the following function?

I would like to find a closed-form expression for the function that is defined as follows: $T_{s}(x) = x^{s}(1 - x^{s}), \text{for prime } x \\ T_{s}(x) = x^{s}, \text{otherwise}$
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### What is $\sum_{i=1}^{n}\frac{F_i}{i}$?

Mathematica is able to evaluate the summation $\sum_{i=1}^{n}\frac{F_i}{i}$ in terms of the Lerch transcendent. It is natural to consider whether or not this summation can be expressed in a more ...
### Closed-from for the series: $\sum_{k=0}^{\infty} \frac{1}{(k!)!}$
As the title says, I'm wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)!}$$ Here I don't mean the double factorial (treated here) ...