Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is essentially a straightforward exercise in complex analysis (integrate on a semicircle in the UHP using the residue theorem, send the radius to $+\infty$ and show that the integral on […]

I have the following equation which is making me problems. $$A_{n} = \sum_{k=0}^{n} \binom{n-k}{k}(-1)^{k}$$ where $n\in\mathbb{N}$. The task is to find a closed form expression for $A_{n}$. I have two questions: What is a closed form expression? We never defined it in my combinatorics class and on the internet there are several different definitions. How […]

By using the integral representation of harmonic numbers and by using elementary integration we can fairly easily find the generating function of squares of harmonic numbers: \begin{equation} {\mathcal S}(x) := \sum\limits_{n=1}^\infty H_n^2 x^n = \frac{Li_2(1)- Li_2(1-x)}{1-x} + \frac{-\log(x) \log(1-x) + [\log(1-x)]^2}{1-x} \end{equation} Now let us take $q \ge 2$. By consecutively dividing both sides by […]

After playing around with transforms of a certain parametric integral, I am inclined to think that the linear combination $$f(n):=\dfrac1{n-2}\left({\,}_2F_1(\dfrac{n-2}{4n},\dfrac12;\dfrac{5n-2}{4n};-1)\right)+\dfrac1{n+2}\left({\,}_2F_1(\dfrac{n+2}{4n},\dfrac12;\dfrac{5n+2}{4n};-1)\right)$$ has a closed form for integer $n$. I know for example that $f(3)=\dfrac{1}{12^{3/4}}\dfrac{\Gamma(\frac14)^2}{\sqrt{\pi}}$. Any ideas? Edit: putting $a:=\frac14-\frac1{2n}$, we can define $g(a):= \frac{8}{1-4a}f(\frac{8}{1-4a})$ to get arguments closer to the “standard” notation used in formula collections. […]

What is closed-form expression for the summation $$ S(n,m)=\sum_{|\alpha|=m} p^{\alpha} = \sum_{\alpha_1 + \cdots + \alpha_n = m} \prod_{i=1}^n p_i^{\alpha_i} $$ as a function of $n$ and $m$? Here $\alpha=(\alpha_1,\cdots,\alpha_n)$ is a $n$-tuple of non-negative integers. For specific values of $n$ the sum has a closed form, e.g. $S(1,m)=p_1^m, S(2,m)=\frac{p_2^{m+1}-p_1^{m+1}}{p_2-p_1}$, etc. I wonder if there […]

I want to calculate: $$\int \frac{1}{x^7 -1} dx$$ Since $\displaystyle \frac{1}{x^7 -1} = – \sum_{i=0}^\infty x^{7i} $, we have $\displaystyle(-x)\sum_{i=0}^\infty \frac{x^{7i}}{7i +1} $. Is there another solution? That is, can this integral be written in terms of elementary functions?

When trying to find a closed form for the expression $$\int x^nf(x)\mathrm{d}x$$ in terms of integrals of $f(x)$ I found that $$\int xf(x)\mathrm{d}x=x\int f(x)\mathrm{d}x-\iint f(x)(\mathrm{d}x)^2$$ $$\int x^2f(x)\mathrm{d}x=x^2\int f(x)\mathrm{d}x-2x\iint f(x)(\mathrm{d}x)^2+2\iiint f(x)(\mathrm{d}x)^3$$ $$\int x^3f(x)\mathrm{d}x=x^3\int f(x)\mathrm{d}x-3x^2\iint f(x)(\mathrm{d}x)^2+6x\iiint f(x)(\mathrm{d}x)^3-6\iiiint f(x)(\mathrm{d}x)^4$$ In general, it seems to be the case that $$\int x^nf(x)\mathrm{d}x=\sum_{k=0}^n(-1)^k\frac{\mathrm{d}^k}{(\mathrm{d}x)^k}x^n\int^{k+1}f(x)(\mathrm{d}x)^{k+1}$$ Using a formula for the repeated derivative of […]

Consider: $$T(n) = 2 T(n-1) + 1$$ with $T(1)$ a positive integer constant $a$. I just stuck in finding a closed form for this simple recursion function. I would appreciate it, if someone gives me a hint.

Assume $k$ positive integer. How can I show that $$ \tag 1 \prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}? $$ I know that $$ \tag 2 \underset{n\geq1}{\prod}\left(1-\frac{k^{2}}{n^{2}}\right)=\frac{\sin\left(\pi k\right)}{\pi k}$$ and so if $k$ is an integer the product is $0$ but how can I use these information?

This question already has an answer here: Evaluate $\sum_{n=1}^\infty nx^{n-1}$ 2 answers

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