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
Filter by
Sorted by
Tagged with
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 $$s(1/2,2) = (1/2)^1+(1/2)^{4}+(1/2)^{9}+(1/2)^{16}+(1/2)^{25}+...$$ ...
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 (...
521 views

336 views

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 $n$. ...
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 ...
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 ?
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 ...
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 ...
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 ...
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 $\dfrac{1}{... 0answers 148 views Summation of$\sum_{n=0}^\infty e^{-\sqrt n}$Is there a closed form for the following sum? $$\sum_{n=0}^\infty e^{-\sqrt n}$$ 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 }$$$\frac{1}{(1+x+x^2+x^3)^3}+\frac{2}{(1+x^2+x^4+x^6)^3}+...+\frac{36}{(1+x^{... 0answers 142 views A closed form for the following Series I was computing some calculations, when I got stuck about a possible closed form for this series:$$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$I proved by hands that it's absolutely convergent by ... 0answers 117 views Infinite Sums which turn out to be Riemann Integrals I'm looking for examples of infinite series which look hard to evaluate at first, but become very simple when viewed as a Riemann integral. An example would be$$\frac{1}{n+1}+\frac{1}{n+2}+ \ldots +\... 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 ... 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 ... 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 ... 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 ... 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. ... 0answers 131 views What is the growth rate of the products of binomial coefficients? Question 1: Are the following empirically observed relationships true $${n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a} \sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg)$$ ... 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): $$\sum_{k=1}^{n-1}\frac{1}{1-\... 2answers 71 views Proving that \sum_{n=0}^r (-1)^n \binom{r}{n} (s+r-n-1)!/(s-n)! = 0 without Taylor expansion Let 0<r\leq s be two integers. I would like to prove that the sum$$\sum_{n=0}^r(-1)^n \binom{r}{n} \frac{(s+r-n-1)!}{(s-n)!}$$is equal to zero. One possible way to prove this is to use the ... 0answers 119 views Find \quad\cot^2(2^{\circ})+\cot^2(6^{\circ})+\cot^2({10}^{\circ})++\cdots +\cot^2({86}^{\circ}) Find$$\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) = \... 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$...
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 ...