# Questions tagged [summation-by-parts]

Summation by parts for discrete variables is the equivalent of integration by parts for continuous variables.

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### Prove $\sum_{k=1}^n\frac{\left(H_k^{(p)}\right)^2}{k^p}=\frac13((H_n^{(p)})^3-H_n^{(3p)})+\sum_{k=1}^n\frac{H_k^{(p)}}{k^{2p}}$

Find $$\sum_{k=1}^n\frac{\left(H_k^{(p)}\right)^2}{k^p}\,,$$ where $H_k^{(p)}=1+\frac1{2^p}+\cdots+\frac1{k^p}$ is the $k$th generalized harmonic number of order $p$. Cornel proved in his book, (...
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### challenging sum $\sum_{k=1}^\infty\frac{H_k^{(2)}}{(2k+1)^2}$

How to prove that \begin{align} \sum_{k=1}^\infty\frac{H_k^{(2)}}{(2k+1)^2}=\frac13\ln^42-2\ln^22\zeta(2)+7\ln2\zeta(3)-\frac{121}{16}\zeta(4)+8\operatorname{Li}_4\left(\frac12\right) \end{align} ...
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### A peculiar Euler sum

I would like a hand in the computation of the following Euler sum (Why isn't here a tag for Euler sums?) $$S=\sum_{m,n\geq 0}\frac{(-1)^{m+n}}{(2m+1)(2n+1)^2(2m+2n+1)} \tag{1}$$ which arises from ...
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### Find the sum of $1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$

Find the sum of $$1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$$ a) $\dfrac{\pi}8(\sqrt2-1)$ b) $\dfrac{\pi}4(\sqrt2-1)$ c) $\dfrac{\pi}8(\sqrt2+1)$ d) $\dfrac{\pi}4(\sqrt2+1)$ ...
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### Study the convergence of the series: $\sum_{n=2}^{\infty} \frac{\cos(nx)\sin(\frac{x}{n})}{\ln n}$

I am studying the convergence of the following series: $$\sum_{n=2}^{\infty}\frac{\cos(nx)\sin\frac{x}{n}}{\ln n}$$ I thought about using the Dirichlet's Test, according to which: If we have a ...
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### How to see that equalities follow from summation by parts?

Let $\{a_n\}$ be a sequence of real numbers. Let $s_n = a_0+...+a_n$. The following equalities appear in a proof I'm reading \begin{align} \sum_{n=0}^\infty a_nx^n &= a_0 + \sum_{n=1}^\infty (s_n ...
I am trying to expand the series $\sum_{k=1}^{n}\binom{n}{k}$ when $n$ is a integer greater then zero, by using summation by parts. I am using the following definition of summation by parts.\begin{...