# Questions tagged [summation]

2,381 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
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 $$s(1/2,2) = (1/2)^1+(1/2)^{4}+(1/2)^{9}+(1/2)^{16}+(1/2)^{25}+...$$ ...
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 ...
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 ...
392 views

224 views

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 ...
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 ...
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 ...
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 ?
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 ...
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 ...
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 ...
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 $\lfloor n\cdot(... 0answers 168 views ### Summation of$\sum_{n=0}^\infty e^{-\sqrt n}$Is there a closed form for the following sum? $$\sum_{n=0}^\infty e^{-\sqrt n}$$ 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 }$$$\frac{1}{(1+x+x^2+x^3)^3}+\frac{2}{(1+x^2+x^4+x^6)^3}+...+\frac{36}{(1+x^{... 0answers 146 views ### A closed form for the following Series I was computing some calculations, when I got stuck about a possible closed form for this series:$$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$I proved by hands that it's absolutely convergent by ... 0answers 120 views ### Infinite Sums which turn out to be Riemann Integrals I'm looking for examples of infinite series which look hard to evaluate at first, but become very simple when viewed as a Riemann integral. An example would be$$\frac{1}{n+1}+\frac{1}{n+2}+ \ldots +\... 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 ... 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 ... 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 ... 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 ... 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. ... 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 ...