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Questions tagged [summation-by-parts]

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

3
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1answer
149 views

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 ...
1
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0answers
57 views

If a series converges, does it still converge if each term is multiplied by an increasing function

Suppose $f: \mathbb{N} \longrightarrow \mathbb{N}$ is a non-decreasing function such that $\sum_{t=1}^{\infty} 2^{-f(t)} < \infty$. Is it possible to find another non-decreasing function $g: \...
2
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0answers
47 views

A rotating polygonal line with increasing side length cannot end up where it started!

Consider a polygonal line $P_0P_1...P_n$ such that $\angle P_0P_1P_2=\angle P_1P_2P_3=...=\angle P_{n-2}P_{n-1}P_{n}$, all measured clockwise. If $P_0P_1>P_1P_2>...>P_{n-1}P_{n}$, $P_0$ and $...
10
votes
1answer
208 views

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 ...
0
votes
3answers
75 views

Prove that $ \sum\limits_{i=1}^ \infty \frac{(-1)^{i+1}}{\sqrt{i}}$ converges, using Abel's summation formula

Let $(a_n)$ and $(b_n)$ be two sequences. Let $A_n = \sum_{k=1}^n a_k$ and $A_0 = 0$. Prove that $$\sum_{n=p}^q a_n b_n = \sum_{n=p}^{q-1} A_n(b_n - b_{n+1}) + A_qb_q - A_{p-1}b_p$$ for each $q\geq p ...
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0answers
53 views

Partial summation formula for two indexes?

We know that partial summation states that given two sequences $(\alpha_n), (\beta_n)$, we can write $$\sum_{n=1}^p\alpha_n\beta_n=\beta_nA_p+\sum_{n=1}^{p-1}(\beta_n-\beta_{n+1})A_n$$ where $A_p=\...
4
votes
5answers
129 views

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{\...
1
vote
1answer
110 views

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 ...
2
votes
2answers
69 views

Sum of $n$ terms of the series: $30+144+420+960+1890+3360+\cdots$

I need to find the general term and the sum of $n$ terms of the series:$$30+144+420+960+1890+3360+\cdots$$ The answer provided my book is: $$U_n=n(n+1)(n+2)(n+4),\quad S_n=\frac{1}{20}n(n+1)(n+2)(n+3)(...
3
votes
3answers
123 views

re-calculating exponential sum when exponent changes

If $A_c$ can be calculated as follows: $A_c=\sum_{k=1}^{N} a_ke^{ck}$ Where c is a known real constant and $a_k$ is a known series comprising real numbers which cannot be described by a function $f(...
2
votes
1answer
55 views

How to show that the following relation is valid for nested summation?

I have seen following relation in a research paper $$\sum_{x_1=1,x_1\neq1}^{K}~\sum_{x_2=x_1+1,x_2\neq 1}^K\cdots \sum_{x_n=x_{n-1}+1,x_n\neq1}^Kf(x_1,x_2,\cdots,x_n)+\sum_{x_1=1,x_1\neq2}^{K}~\sum_{...
0
votes
2answers
45 views

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 ...
2
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1answer
62 views

Using summation by parts on a combination

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{...
1
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0answers
39 views

An identity via summation by parts

In a text I am currently reading, the authors use through their computations the following identity : $$\sum_{i=1}^n \left( a_{i+1}-a_i-\frac{1}{n}\left(a_{i+n}-a_i\right)\right) v_i = \frac{1}{n}\...
-2
votes
1answer
204 views

simplification of double summation [closed]

I have solved double summation problem,.Kindly check it whether it is correct or not?? $$\sum_{j=1}^3\sum_{i=1}^j (i+j) = 12$$ thanks
6
votes
2answers
254 views

If for any $k$, $\sum\limits_{n=0}^\infty a_n^k=\sum\limits_{n=0}^\infty b_n^k$, then$(a_n)=(b_{σ(n)}),\ σ \in{\mathfrak S}_{\mathbb N}$

Let $(a_n)_{n≥0}$ and $(b_n)_{n≥0}$ be two sequences of a nomed algebra such that $\sum{\| a_n\|}$ and $\sum{\| b_n\|}$ converge, and$ $$$\forall n, \ a_n, b_n \neq 0$$ Show that $(\forall k \in \...
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0answers
61 views

Number Theoretic Estimate $\sum_{p\leq y}\frac{\log p}{p}h(pt)=\int_{t}^{yt}\frac{h(v)}{v}\ dv+O\big(h(y)\big)$

I'm working through a paper and the author states the following result with little justification, please check my working. Let $h:[y,\infty)\to\mathbb{R}^{>0}$ be decreasing with a continuous ...
1
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1answer
63 views

$\sum a_nb_n$ convergence theorem (Rudin)

Why does the first inequality (in the chain) hold? Is $$\left\lvert \sum_{n=p}^{q-1}A_n(b_n - b_{n+1})\right\lvert \le M \left\lvert \sum_{n=p}^{q-1}(b_n - b_{n+1})\right\lvert$$ true? and why?
13
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2answers
296 views

On twisted Euler sums

An interesting investigation started here and it showed that $$ \sum_{k\geq 1}\left(\zeta(m)-H_{k}^{(m)}\right)^2 $$ has a closed form in terms of values of the Riemann $\zeta$ function for any ...
4
votes
1answer
128 views

Estimating sums over primes

I’m interested in estimating the sum $$ \sum_p \pi\left(\frac{x}{p^3}\right) $$ where the sum is over all primes $p$. (It’s a finite sum of course, you can cut off at $\sqrt[3]{x/2}$.) The goal is a ...
14
votes
1answer
442 views

Can we have a power density but not a natural density?

For $M \subset \mathbb{N}$ (in this post I follow the convention $\min \mathbb{N} = 1$) and $\alpha \in [0,1]$ define $$S_{M,\alpha}(x) = \sum_{\substack{n\in M \\ n \leqslant x}} \frac{1}{n^{\alpha}}...
0
votes
2answers
736 views

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 ...
1
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0answers
43 views

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 $M<N$.
1
vote
1answer
35 views

Represent a series into a loop.

I got this series $S= (1^2 ) - (2^2 ) + (3^2 ) - (4^2 )+...+(-1)^{n+1 }* (n^2 )$ ...
0
votes
1answer
14 views

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 $k=0$ ...
1
vote
1answer
75 views

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: $$ \sum_{y < n \leq x} f(n) = \int_y^x{f(t)dt} + \int_y^x(t - \lfloor t \rfloor)f'(t)dt +f(x)(\lfloor x \rfloor - x) -f(y)(\lfloor y \rfloor -y)$$ (...
1
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0answers
110 views

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 ...
86
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4answers
3k views

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 ...
2
votes
1answer
264 views

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, ...
5
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2answers
701 views

Striking applications of summation by parts

In the same vein as this question Striking applications of integration by parts I'd also like to have a list of some good applications of the discrete version: summation by parts.
6
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2answers
406 views

Summation by Parts

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 =...