Questions tagged [summation]

Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

2,381 questions with no upvoted or accepted answers
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26
votes
0answers
880 views

A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. I'm looking at the (gap-)series $$ s(1/2,2) = (1/2)^1+(1/2)^{4}+(1/2)^{9}+(1/2)^{16}+(1/2)^{25}+... $$ ...
23
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0answers
981 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers (for ...
22
votes
0answers
446 views

Geometric representation of Euler-Maclaurin Summation Formula

In Tom Apostol's expository article (here's a free link), upon seeing the figure below (or this from the Wolfram project) I was expecting more diagrams to come to continue the error decomposition of ...
22
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0answers
392 views

A combinatorial identity involving generalized harmonic numbers

[This problem has now an answer here.] The $n$-th harmonic number is defined as $$ H_n=\sum_{k=1}^{n}\frac{1}{k}, $$ and the generalized harmonic numbers are defined by $$ H_{n}^{(r)}=\sum_{k=1}^{n}\...
17
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0answers
406 views

Convergence acceleration technique for $\zeta(4)$ (or $\eta(4)$) via creative telescoping?

Question Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar ...
14
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0answers
300 views

On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$

Motivation $\newcommand{Li}{\operatorname{Li}}$ It is already known that: $$\Re\Li_2(1+i)=\frac{\pi^2}{16}$$ $$\Re\Li_3(1+i)=\frac{\pi^2\ln2}{32}+\frac{35}{64}\zeta(3)$$ And by this question, ...
14
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0answers
956 views

How to find the approximate basic period or GCD of a list of numbers?

I want to tell the number which act as the best approximate basic period (or wavelenght as pointed out by Eric) of a list of real numbers: e.g for {14, 21, 35} we should obtain 7 as the basic period, ...
13
votes
3answers
293 views

Basel Problem - Area of $\frac 16$ of Circle with Radius $\sqrt{\pi}$.

There are several proofs to the solution of the well-known Basel Problem, i.e. $$\sum_{n=1}^\infty \frac 1{n^2}=\frac {\pi^2}6$$ Is is possible to create a geometrical interpretation of this ...
12
votes
1answer
392 views

Double factorial as a sum

I believe the following equality to hold for all integer $l\geq 1$ $$(2l+1)!2^l\sum_{k=0}^l\frac{(-1)^k(l-k)!}{k!(2l-2k+1)!4^k}=(-1)^l(2l-1)!!$$ (it's correct for at least $l=1,2,3,4$), but cannot ...
11
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0answers
206 views

Triple sum $\sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} \sum\limits_{c=1}^{\infty} \frac{\cos a \cos b \cos c}{a^2 + b^2 + c^2}$

We have poor water heating system in our countryside house (currently it takes 4 hours to warm up the water), and my father has decided to improve it; he bought a water tank and placed it up in the ...
11
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0answers
219 views

Are $(2,28)$ and $(5,3207)$ the only solutions $(m,n)\in\mathbb{N}^2$?

I noticed something as I was playing around with prime numbers. By denoting $p_i$ the $i^{\text{th}}$ prime number, I discovered the following: $$ \begin{align}\prod_{i=1}^2\left(p_i^{ \ 2}+i\right)&...
10
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0answers
224 views

Finite analog of these two infinite series with inverse tangents?

The identity $$ \sum_{n=1}^{\infty}\chi(n)\arctan e^{-\alpha n}+\sum_{n=1}^{\infty}\chi(n)\arctan e^{-\beta n}=\frac{\pi}{8}, \qquad \alpha\beta=\frac{\pi^2}{4},\tag{1} $$ where $\chi(n)=\sin\frac{\...
10
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0answers
159 views

Could this conjecture be proved ? (sum of even powers of cotangents in arithmetic progression )

Having tried (in vain) to answer this question, I worked the explicit formulae of $$\color{blue}{S_k=\sum _{n=1}^m \Big[\cot \left(\frac{n \,\pi }{2 m+1}\right)\Big]^k}$$ where $k$ is an even integer....
10
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0answers
329 views

Closed form for $\sum _{k=0}^n (-1)^k \binom{n}{k} \binom{n+k}{n} (H_n-H_k) x^k$

The sum originated from this question and I found (experimentally) two closed form cases: $$\sum _{k=0}^n (-1)^k \binom{n}{k} \binom{n+k}{n} (H_n-H_k)=(-1)^{n+1} H_n$$ $$\sum _{k=0}^n (-1)^k \binom{n}{...
10
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0answers
130 views

Can a prime have arbitary many representations as a sum of two perfect powers?

Let $p$ be a prime number and $f(p)$ be the number of representations $$p=a+b$$ with perfect powers $0<a< b$ For example, $13$ has the only representation $$4+9=13$$ hence $f(13)=1$ $$41=9+32=...
10
votes
1answer
236 views

Considering the equation, $6 + (2k+1)\sum_{n=1}^{2k+1}p_n^{ \ \ 3}(-1)^{n+1} = x^2$.

I noticed that, $$\begin{align}3(2^3 - 3^3 + 5^3) + 6 &= 18^2 \\ \text{and } \qquad 5(2^3 - 3^3 + 5^3 - 7^3 + 11^3) + 6 &= 74^2.\end{align}$$ These equations are of the form, $$6 + (2k+1)\sum_{...
9
votes
1answer
379 views

How to estimate a specific infinite sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...
8
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0answers
166 views

Is $2^9$ the only power of two that is the sum of two odd perfect powers?

Let $m\ge 1$ be an integer. For which $m$ can we find odd perfect powers $a,b$ with $a+b=2^m$ ? The only solution, I found for $m\le 80$ is $m=9$ with the representation $$2^9=13^2+7^3$$ The ...
8
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0answers
356 views

Is there a closed form for $\,_4 F_3(1,1,1,3; 3/2,5/2,5/2;1)$?

A semi-algebraic generalization of the Steiner surface has appeared, $$S = \left\{(x,y,z,t) \space \vert \space t^2(1-x^2-y^2-z^2-t^2) - (x^2 y^2 + x^2 z^2 + y^2 z^2 - 2 x y z) \geq 0 \right\}$$ ...
8
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0answers
115 views

Extending Bell Numbers to Fractional Values

An identity of the Bell numbers is given by $$B_n=\frac{1}{e}\sum_{x=1}^\infty \frac{x^n}{x!}$$ and I was wondering if it would be valid to define fractional Bell numbers in the same way, to preserve ...
8
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0answers
287 views

How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done with this series: $$ \sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$ As hardmath says in his comment, the series ...
7
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0answers
189 views

Sum over invertible 0-1 matrices

I stumbled across the following formula when working on a research problem in theoretical computer science. I checked its correctness up to $N=5$ with a computer. I am looking for a simple proof of it....
7
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0answers
99 views

Sum of numbers $x+y$ satisfying $x^2+y^2=p$ with $p$ being a prime number

(This question has been inspired by a similar one from James Johnson) It's a well known fact that every prime number $p$ such that: $$p\equiv 1 \pmod 4\tag{1}$$ can be represented as a sum of two ...
7
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0answers
126 views

Prove that sums of powers are equal

All numbers between $1$ and $10^{20}$ ($10^{20}$ not included) are divided into two sets: one with numbers, whose sum of digits is odd and the other with remaining numbers(those, whose sum of digits ...
7
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0answers
693 views

Closed Form for this Taylor Series?

Does anybody know whether or not this sum has a closed form? $$f(x)=\sum_{n=0}^\infty \frac{x^n}{n!(2^n+1)}$$ I can't get WA to even understand it when I type it in. For context, the reason I want to ...
7
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0answers
297 views

Is there a way to write this recurrence relation in faster-to-program manner?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
7
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0answers
215 views

Finite sum with three binomial coefficients

I need to find a closed form expression, if there is one, of the following sum: $$\sum_{j=0}^m{n+1-k\choose j}{k-1\choose m-j}{A+2-k+m-j\choose m-j+2}$$ where all parameters are integers, $~1\leq k\...
7
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0answers
189 views

Double sum involving $\cos$

I ran across a double sum and was wondering if someone may be adept at evaluating it. I must admit that my double summation skills could be better, and I am always ready to learn more. Show that: $$\...
7
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0answers
398 views

Can we interchange the Integral and Summation when a limit is $\infty$?

I was trying to Evaluate the Integral: $$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$ $$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot \frac{1}{1+...
7
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0answers
339 views

How find this sum $\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$

Question: Find the sum $$I=\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$$ where $$L_{n}(k)=1-\dfrac{1}{2^k}+\dfrac{1}{3^k}-\cdots+\dfrac{(-1)^{n-1}}{n^k}$$ since $$L_{n}(2)=1-\dfrac{1}{2^2}+\...
7
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1answer
242 views

How to calculate probability of these two pair-of-sums ($S_{n}$ and $T_{n}$) and ($SEvenF_{n}$ and $SOddF_{n}$) being the same?

Say we have a sequence of $n$ positive integers, we can assume they're randomly chosen, let's call it $U_{n}$. Let $S_{n}$ = sum of $U_{n}$ from $1$ to $n$. Let $T_{n}$ = sum of $n$ from $1$ to $n$. ...
6
votes
1answer
56 views

Sum of multinomial coefficients with bounded indices

I want to know if there's a closed formula for $$ f_{k}(n_{1}, \dots, n_{k}) = \sum_{i_{1} = 0}^{n_{1}} \cdots \sum_{i_{k} = 0}^{n_{k}} \frac{(i_{1} + \cdots + i_{k})!}{i_{1}!\cdots i_{k}!} $$ for $k\...
6
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0answers
112 views

Value of $\sqrt{\sum_{k=1}^{1}k+\sqrt{\sum_{k=1}^{2}k+\sqrt{\sum_{k=1}^{3}k…}}}$

Note: I would like to add this is very similar to this problem posted a couple months ago, which is the inspiration for this question, was proven to converge in one of the answers, and I believe a ...
6
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0answers
162 views

What is the growth rate of the products of binomial coefficients?

Claim: Experimental data seems to suggest that $$ {n \choose 1^a b}{n \choose 2^a b}{n \choose 3^a b}\cdots {n \choose m^a b} \sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{ab+3b}\bigg) $$ where $a$ and ...
6
votes
0answers
120 views

Is $n=13$ the only solution to this: $\pi\left(\sum_{i=1}^n\pi(i)\right)-1=n$.

I was messing around with the prime counting function $\pi(n)$ because I was bored, but then I noticed something. The equation $$\pi\left(\sum_{i=1}^{n}\pi(i)\right)-1=n,$$ has the ...
6
votes
0answers
130 views

Is there another prime $p$ such that $S(p)$ is prime?

Denote $$S(p):=2^2+3^3+5^5+\cdots +p^p$$ $S(p)$ is prime for $p=3,7,89$. Is there another prime $p$ such that $S(p)$ is prime ? Is the number of primes $p$ such that $S(p)$ is prime, finite ?
6
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0answers
238 views

Alernative way of solving summation $5+55+555+\ldots$

Question Find the summation for the series-: $$5+55+555+5555+...$$ I know it is a duplicate of this, but still, I am posting this because i was thinking of solving another way due to which I got ...
6
votes
0answers
211 views

Fractional part summation and asymptotic expansion error

Let us consider the sum $$\displaystyle T_K=\sum_{n \geq \sqrt{K}}^{{K}} \left\{ \sqrt {n^2-K} \right\} $$ where $K$ is a positive integer and where $\{ \}$ indicates the fractional part. Its ...
6
votes
0answers
321 views

Finding a Formula for a Sum

Given integers $1\le j\le n$, let $p$ denote the largest prime at most $n$. I want to sum $$1/i$$ over all $i=2^{a_2}3^{a_3}\cdots p^{a_p}$ $\,(a_l\ge 0)$ such that both $j,n$ have at least 2 more ...
6
votes
0answers
97 views

Equality of Floors of some Partial Sums

Let $S_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}$ denote the $(n+1)^{st}$ partial sum in the series expansion for $e=\sum_{k\ge 0}\frac{1}{k!}$. I want to prove that $\lfloor n\cdot(...
6
votes
0answers
168 views

Summation of $\sum_{n=0}^\infty e^{-\sqrt n}$

Is there a closed form for the following sum? $$\sum_{n=0}^\infty e^{-\sqrt n}$$
6
votes
0answers
138 views

Help with the following summation when $x^{37}=1,x\neq 1$

I want to find the following summation $\text{Let }x^{37} = 1 \text{ and } x \neq 1,$ $\\ \text{Find the summation of }$ $$\frac{1}{(1+x+x^2+x^3)^3}+\frac{2}{(1+x^2+x^4+x^6)^3}+...+\frac{36}{(1+x^{...
6
votes
0answers
146 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
6
votes
0answers
120 views

Infinite Sums which turn out to be Riemann Integrals

I'm looking for examples of infinite series which look hard to evaluate at first, but become very simple when viewed as a Riemann integral. An example would be $$\frac{1}{n+1}+\frac{1}{n+2}+ \ldots +\...
6
votes
0answers
366 views

Subset Sum Problem Variation?

There are $100$ cards with a unique number from $1$ to $100$ written over them. How many ways can someone pick exactly $5$ cards where the numbers on them sum to $100$? I am not sure but this could ...
6
votes
0answers
451 views

Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
6
votes
0answers
82 views

Series with $k^2$ coefficients

Let $\{S_k\}$ be a sequence of numbers. Then by reversing the sum we have the following, $$A=\sum_{k=0}^n k\cdot S_k \cdot S_{n-k} = \sum_{k=0}^n (n-k) \cdot S_{n-k} \cdot S_{k} .$$ Thus adding these ...
6
votes
0answers
807 views

Proof for an identity involving a sum of binomial coefficients

I am moving through a On The Average Height of Planted Plane Trees by Knuth, de Bruijn and Rice, 1972), and I would like to trade a weaker result for simpler mathematical tools, because my skills are ...
6
votes
1answer
217 views

Tricky Sum involving Binomial Coefficients and Sine

I am stumped by the sum $$\sum_{x=0}^n \binom{n}{x}\sin\big(\frac{\pi x}{n}\big)$$ but I can't figure it out. I tried expanding the taylor series of sine and using Euler's identity, but to no avail. ...
5
votes
0answers
111 views

Flint Hills series $\sum_{n=1}^{\infty}\frac{1}{n^3\sin^2 n}$

The well known Flint Hills series, $$\sum_{n=1}^{\infty}\frac{1}{n^3\sin^2 n}$$Since I'm aware of the fact that the series convergence issue is still unknown or falls under unsolved problem ...

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