# Questions tagged [summation]

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

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### 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 $$s(1/2,2) = (1/2)^1+(1/2)^{4}+(1/2)^{9}+(1/2)^{16}+(1/2)^{25}+...$$ ...
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### 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 (...
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### 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 $n$. ...
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### 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 ...
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### 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 ?
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### 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 ...
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### 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 ...
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### 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 ...