Questions tagged [summation-by-parts]

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

31 questions
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challenging sum $\sum_{k=1}^\infty\frac{H_k^{(2)}}{(2k+1)^2}$

where $H_n^{(m)}=1+\frac1{2^m}+\frac1{3^m}+...+\frac1{n^m}$ is the $n$th harmonic number of order $m$. this problem was proposed by Cornel Valean on his FB page. I tried to solve it using ...
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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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Summation of series by method of differences

1.5.9+2.6.10+3.7.11 ... n terms N th term would be n(n+4)(n+8) My try N th term =n[(n+1)+3)][n+2+(6)] I am unable to break the series further . How can I simplify this further using method of ...
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A harmonic sum with error function

I need an approximation or a closed form for the series $$\sum_{a=1}^{M} \frac{1}{a} \, \text{erf} \left(\frac{M}{N} \, a \right)$$ where $M<N$.
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Represent a series into a loop.

I got this series $S= (1^2 ) - (2^2 ) + (3^2 ) - (4^2 )+...+(-1)^{n+1 }* (n^2 )$ ...
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Separate zero-th degree terms

I have following triple summation expression: $$\sum_{i=0}^{m}\sum_{j=0}^{i}\sum_{k=0}^{n}a_{i,j,k}x^{i-j+k}.$$ I want to separate terms with $0$-th degree of $x$ from others. I understand that $k=0$ ...
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Why do the first two terms of Euler's summation by parts formula not cancel each other out?

Euler's summation by parts formula states that: $$\sum_{y < n \leq x} f(n) = \int_y^x{f(t)dt} + \int_y^x(t - \lfloor t \rfloor)f'(t)dt +f(x)(\lfloor x \rfloor - x) -f(y)(\lfloor y \rfloor -y)$$ (...
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Improper integral $\int_{1}^{\infty}\frac{\left \{ x \right \}-\frac{1}{2}}{x(\log x+z)}dx$

We have the integral : $$\int_{1}^{\infty}\frac{\left \{ x \right \}-\frac{1}{2}}{x(\log x+z)}dx$$ $\left \{ x \right \}=$ fractional part of $x$, and $z$ is a complex variable whose real part is ...
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A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent

As the title says, I would like to launch a community project for proving that the series $$\sum_{n\geq 1}\frac{\sin(2^n)}{n}$$ is convergent. An extensive list of considerations follows. The ...
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Summation by parts interpretation

So in Rudin, I'm given the theorem for summation by parts: $$\text{Given two sequences} \{a_n\}, \{b_n\}, \text{put}\\ A_n = \sum_{k=0}^n a_k\\$$ $\text{if}$ $$n \ge 0; \text{put}\ A_{-1} = 0$$ Then, ...
I am trying to solve $\sum\limits_{k=1}^n\frac{2k+1}{k(k+1)}$ using summation by parts: $\sum u\Delta v=uv-\sum Ev\Delta u$ $u = 2x+1, \Delta v=1/x(x+1)=(x-1)_{-2}$ $v=-(x-1)_{-1}=-1/x$ \$\Delta u =...