Questions tagged [summation]

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

2,044 questions with no upvoted or accepted answers
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21
votes
0answers
674 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 $$ s(1/2,2) = (1/2)^1+(1/2)^{4}+(1/2)^{9}+(1/2)^{16}+(1/2)^{25}+... $$ ...
21
votes
0answers
940 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 (...
19
votes
0answers
521 views

Prove a strong inequality $\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2-\frac{7\ln 2}{8\ln n}\right)\sum_{k=1}^n\frac 1{a_k}$

For $a_i>0$ ($i=1,2,\dots,n$), $n\ge 3$, prove that $$\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2\color{red}{-\frac{7\ln 2}{8\ln n}}\right)\sum_{k=1}^n\frac 1{a_k}.$$ The case without $\...
17
votes
0answers
317 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}\...
16
votes
0answers
374 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 ...
15
votes
0answers
296 views

Geometric representation of Euler-Maclaurin Summation Formula

When reading Tom Apostol's expository article (or the free link), I was expecting more diagrams to come that follow the figure below (or this from the Wolfram project). It was a disappointment not ...
13
votes
0answers
756 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, ...
12
votes
3answers
251 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 ...
11
votes
0answers
206 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)&...
11
votes
1answer
354 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 ...
10
votes
0answers
119 views

On an asymptotic improvement of AMM problem 11145 (April 2005)

Motivation Motivated by this question, I tried improve the inequality $$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}\le2\sum_{k=1}^{n}\dfrac{1}{a_{k}}$$ asymptotically. In other words, with ...
10
votes
0answers
297 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{...
10
votes
1answer
227 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
0answers
171 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 ...
9
votes
0answers
118 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=...
9
votes
1answer
361 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
votes
0answers
99 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
votes
2answers
187 views

Combinatorial sum identity for a choose function $\sum\limits_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$

I want to show that the following holds: $$\sum_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}.$$ I have an idea of what is going on here. On the RHS we are selecting $r+s+1$ ...
8
votes
0answers
272 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
votes
0answers
156 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 ...
7
votes
0answers
326 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, $$S = \left\{(x,y,z,t) \space \vert \space t^2(1-x^2-y^2-z^2-t^2) - (x^2 y^2 + x^2 z^2 + y^2 z^2 - 2 x y z) \geq 0 \right\}$$ ...
7
votes
0answers
460 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
votes
0answers
293 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
votes
0answers
195 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
votes
0answers
162 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
votes
1answer
179 views

Summation of a function

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}...p_k^{c_k}$ where $c_k = ...
7
votes
0answers
336 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
votes
0answers
334 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
votes
1answer
226 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
0answers
118 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
128 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
votes
0answers
186 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
187 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
280 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
1answer
109 views

Prove the Result is 1/2

Given a natural number $n>1$, add up all the fractions $\dfrac{1}{pq}$ , where $p$ and $q$ are relatively prime, $0< p < q \le n$, and $p+q>n$. Prove that the result is always $\dfrac{1}{...
6
votes
0answers
148 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
128 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
142 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
117 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
362 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
411 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
81 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
800 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
140 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
131 views

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

Question 1: Are the following empirically observed relationships true $$ {n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a} \sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg) $$ ...
5
votes
0answers
51 views

Sum involving roots of unity: $\sum \frac{1}{1-\omega^k}$

Let $\omega_n^k=\exp(2\pi ik/n)$ be the $n$-th roots of unity. I've come across the following sum a couple of times now (for example in a problem on hydrodynamics): $$ \sum_{k=1}^{n-1}\frac{1}{1-\...
5
votes
2answers
71 views

Proving that $\sum_{n=0}^r (-1)^n \binom{r}{n} (s+r-n-1)!/(s-n)! = 0$ without Taylor expansion

Let $0<r\leq s$ be two integers. I would like to prove that the sum $$\sum_{n=0}^r(-1)^n \binom{r}{n} \frac{(s+r-n-1)!}{(s-n)!}$$ is equal to zero. One possible way to prove this is to use the ...
5
votes
0answers
119 views

Find $\quad\cot^2(2^{\circ})+\cot^2(6^{\circ})+\cot^2({10}^{\circ})++\cdots +\cot^2({86}^{\circ})$

Find $$\cot^2(2^{\circ})+\cot^2(6^{\circ})+\cot^2({10}^{\circ})++\cdots +\cot^2({86}^{\circ})$$ $\mathbf {My Attempt}$ I tried to write the sum backward like this $$S=\sum_{n=1}^{22} \cot^2(4n-2) = \...
5
votes
0answers
116 views

On the integral $\int_1^\infty\big(\{x\}^n-\frac1{n+1}\big)\frac{dx}x$

According to Dirichlet's test (integral version), $$ I_n=\int_1^\infty\big(\{x\}^n-\frac1{n+1}\big)\frac{dx}x $$ converges, where $n$ is a positive integer and $\{x\}$ denotes the fractional part of $...
5
votes
1answer
135 views

Sum of 2 raised to the power of every element in a line of Pascal’s triangle

I know that the elements of a line in Pascal’s triangle add up to $2^n$ . What about: $$\sum_{k=0}^n 2^{\binom{n}{k}}$$ For example, line $n = 2$ adds up to $8$. $n = 3$ adds up to $20$. Is there any ...