# 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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### How to show $\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{H_{n-1}}{n^2(n+k)^2}=\sum_{n=1}^\infty\frac{H_{n-1}^{(2)}}{n^3}$ using series manipulation?

Using integration, I managed to show that $$\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{H_{n-1}}{n^2(n+k)^2}=\sum_{n=1}^\infty\frac{H_{n-1}^{(2)}}{n^3}\tag1$$ But I would like to prove the equality using ...
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### Double Summation indexes problem

I have the following sum: \begin{equation} \sum_{j=0}^{a} \sum_{k=0}^{n-2j} c_{jk}\,\, x^{\,j+k} \end{equation} Where $a=\lfloor n/2\rfloor$. I want to convert the previous sum to other like: \...
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### Proof of Expected value by Abel's Summation

Using Abel's summation lemma, prove that: $$\mathbb{E}[X] = \int_0^\infty (1-F(x)) \ dx,$$ where $F(x)$ is the CDF of a random variable $X$ I do know how to prove this equation by another ...
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### Rearrange a Triple Summation with constraints

I have the following triple summation: \begin{equation} \sum_{m=0}^{m_0}\sum_{j=0}^m \sum_{k=0}^{2m_0-2j} a_{kjm} x^{2(j+k)} \end{equation} I think I should be able to simplify it to ...
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### Summation with fractions, discrete calculus

Here is the summation question $$\sum_{k=1}^{n-1}k\left(1+\frac{1}{2}+\dots+\frac{1}{k}\right)$$ I think it should be solved by the technique of discrete calculus (summation by parts). Can someone ...
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### a question on Bernoulli function in the book of Tenenbaum

In section 0.2 of Introduction to Analytic and Probabilistic Number Theory by Gérald Tenenbaum, I read that "One easily verifies that these assumptions imply the identity...". I started from the left ...
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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 =...