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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283 views

Summation of the series $s_b(p)=\sum_k b^{k^p}$ by a double sum in a sense like Ramanujan-method

From some older context I am re-considering the following variant of the geometric series $$ s_b(p)=\sum_{k=1}^\infty b^{k^p} $$ for the convergent cases $0 \lt b \lt 1$ and $ 0 \lt p$ first. I'm ...
3
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70 views

Sum $\sum_{n\geqslant 1} \frac{(-1)^n \sin(n\theta)}{n^3}$ using complex analysis

The following question appeared on an old complex analysis exam. (The only tools we have learned are simple series for $\sin$, $\cos$, $\exp$ etc., the residue theorem and some tools from real ...
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68 views

Is there any closed expression for $\sum_{k=0}^\infty r^{k^2}$?

Let $r$ be any real number with $0 < r < 1$. Then, of course, there is a closed expression for $$\sum_{k=0}^\infty r^k = \frac{1}{1-r}$$ But does there also exist a closed expression for $$\sum_{...
3
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57 views

How to prove the limit related to the following infinite series?

I would appreciate it if you can tell me how to prove this limit $$\lim_{x\to 0^{+}}\sum_{n=2}^{\infty}\frac{(-1)^{n}}{(\log n)^{x}}=\frac{1}{2}$$
3
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70 views

Finite double sum $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$; any advanced summation technique?

Let $M,N,c$ be positive integer. It was astonishing when trying to solve $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$ to obtain this rather complex looking result \begin{align*} ...
3
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96 views

Limit of an alternating series

Given some fixed integer $n$ (but one may take $n = 0$ for simplicity), I would like to compute the following limit: $$\lim_{N \to \infty}\sum_{\substack{|u_1|, |u_2|, |u_3|, |u_4| \leq N\\|u_1 + u_2 ...
3
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73 views

Simplifying an infinite sum

Question: The sum of $$1-\frac16+\frac16\times\frac14-\frac16\times\frac14\times\frac{5}{18}+\cdots$$ is: A) $\frac23$ B)$\frac{2}{\sqrt3}$ C)$\sqrt\frac23$ D)$\frac{\sqrt3}{2}$ After looking ...
3
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66 views

Computing closed form of $\sum_{k=0}^n b^{\alpha^k}$

Let $$\sum_{k=0}^n b^{\alpha^{k}}$$ be a sum where $b \in \mathbb N_+$ and $\alpha \in \mathbb R, \alpha > 1$. What is its name and how can I calculate its closed form? \begin{align} \sum_{k=0}^n ...
3
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68 views

Replacing $n!$ with Stirling's approximation in $e^x = \sum_n \frac{x^n}{n!}$

I was wondering if there is a closed-form expression for $$\sum_{n=0}^{\infty} \frac{x^n}{e^{-n}n^n},$$ although I expect there is none because Mathematica cannot compute it. However, from Stirling'...
3
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49 views

Analyitical solution to summation

Suppose $A$, $B$ and $C$ are positive constants with $A,B,C \in (0,1)$. I have the following summation: $1+A(1-BC)+A^2(1-BC)(1-B^2C)+A^3(1-BC)(1-B^2C)(1-B^3C)+...$ $\Rightarrow \sum_{i=0}^{\infty}A^...
3
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86 views

Combinatorial proof of $ \sum_{k=0}^{n}\frac{1}{\binom{n}{k}} = \frac{n+1}{2^{n+1}}\sum_{k=0}^{n}\frac{2^{k+1}}{k+1}$

A recurrence relation of $$S_n =\sum_{k=0}^{n} \frac{1}{\binom{n}{k}}$$ is $$ \frac{n+2}{\binom{n}{k}} - \frac{2n+2}{\binom{n+1}{k}} = \frac{n-k}{\binom{n}{k+1}} - \frac{n+1-k}{\binom{n}{k}}, \quad 0 ...
3
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73 views

How can I evaluate the below mentioned series without using a computation software?

I have been trying to evaluate $\displaystyle\sum_{m=0}^{2^{2^5}-1}\frac{2}{\prod_{n=1}^5\bigl((m+2)^{\frac{2}{n}}+(m)^{\frac{2}{n}}\bigr)}$ for quite a long time. I tried various approaches but ...
3
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143 views

Which natural numbers are the sum of three positive perfect squares?

In this question : Which positive integers $n$ can be expressed as $n=a^2+b^2+c^2$ , but not with positive $a,b,c\ $? I asked for the classification of the natural numbers being the sum of three ...
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137 views

Which positive integers $n$ can be expressed as $n=a^2+b^2+c^2$ , but not with positive $a,b,c\ $?

Let $S$ be the set of positive integers $n$, such that the equation $$a^2+b^2+c^2=n$$ has a solution in non-negative integers but not in positive integers. Since $n\in S$ if and only if $4n\in S$, we ...
3
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86 views

A conjecture regarding a strange infinite sum

Numerical calculations indicate that the following conjecture is true $$ \sum _{n=0}^{\infty} \frac{\prod _{k=0}^{n-1} \left(x+e^{-k}\right)}{\prod _{k=0}^n \left(1+te^{-k}\right)}t^n =\frac{1}{1-xt}.\...
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71 views

Finding upper and lower sum of a Heaviside function.

I have been asked to find the lower and upper sum of a Heaviside function. It is a combination of a Heaviside function, like: H(x-1) + (3x-12)H(x+1). I have been able to draw the graph for this, ...
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862 views

Sum of minima is at most minimum of sum

I'd like to show the conditions under which the two are equal? $$\min_{x_i,y_j}\sum_{ij} |f(x_i,y_j)|^2 \ge \sum_{ij} \min_{x_i,y_j}|f(x_i,y_j)|^2$$ I believe the inequality can be proved as follows:...
3
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66 views

Computing $\sum_{i = 0}^n \frac{n!}{i!} \mod p$

I'm trying to find the fastest way to compute $$\sum_{i = 0}^n \frac{n!}{i!} \mod p$$ where one can suppose that $p$ is a prime number. Given that $n$ can be a huge number, the only simple way I can ...
3
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71 views

An interesting extension of a combinatorics problem

Yesterday , i saw a question on this site.. I don't know where it went now.. It was as follows Q. Find the number of ways of going from (0,0) to (m,n) given that one step can be :- m,n>0 either 1 ...
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64 views

Vandermonde-like identity mixing combination and permutation

Do you know any identity that could reduce the following quantity: $\sum \limits_{k=0}^{min(p,n)}\frac{n!}{(n-k)!}\binom{p}{k}$ It looks like Vandermonde identity where addends are multiplied by $k!$...
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60 views

Index of summation for a product

In the expression $$\sum_{k=1}^nk=\frac12n(n+1)$$ $k$ is called the index of summation. What is $j$ called in $$\prod_{j=1}^nj=n!$$ I have seen web pages like https://math.illinoisstate.edu/day/...
3
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156 views

Fast Summation over $\frac{n^2}{k^2}$

How to find $\sum\limits_{k=1}^n \lfloor\frac{n^2}{k^2}\rfloor$ where n is of order $10^{18}$? I have been able to calculate it for n of order $10^{14}$ by using fast-summation, but it does not seem ...
3
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74 views

Series $\sum\limits_{n=0}^{\infty} \frac{(-1)^n}{(3n + k)!}x^{3n + k}, \ k = 0, 1, 2$

Everyone knows that $e^{i\theta} = \cos \theta + i \sin \theta$ for any complex argument $\theta$, and that the McLaren series expansion of say $\sin(z)$ is given by: $$ \sin(x) = \sum\limits_{n = 0}^...
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54 views

Flipping the order of an infinite series and assigning it a value?

I was recently wondering if there was some method to "flip" the series and still assign it a value for divergent series. To illustrate what I mean: $$ S(x,n) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n ...
3
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128 views

Double summation of binomial probabilities

I want to evaluate the following summation: $$\sum^{n-1}_{k=0}{n\choose k}x^k(1-x)^{n-k-1}\sum^{n-1}_{j=k}{n-1\choose j}y^j(1-y)^{n-j-1}.$$ Both $x$ and $y$ are some numbers in $[0,1]$. Each one is ...
3
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54 views

Finite rational sum

By a direct calculation, it is easy to deduce that the simple finite sum $$\sum\limits_{j = 1}^n {\frac{{{r^2}}} {{{j^2}r + j{r^2}}}} = {H_n} + {H_r} - {H_{n + r}},$$ where $H_n$ denotes the harmonic ...
3
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63 views

Evaluate a finite sum in a large N limit

I have a finite sum which I evaluate, and subsequently take a limit to retain only part of the result. I am wondering if it is possible to apply this limit earlier to simplify the finite sum. The ...
3
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101 views

bound sum with parameter , revisited

I need to prove (to be more precise: would like to understand why) that it is sufficient for $g$ to be $|g(x)|\le \frac{c}{(1+|s|)^\gamma}$ with $\gamma > 1$ in order for $$\beta(s) = \sup_{0\le ...
3
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277 views

$f(8) \geq 1$ and $f(n)\geq 2f(\lceil \frac n2-n^{2/3} \rceil)$. Can we deduce $\exists C>0: f(cn) \geq n$?

Let $f : \Bbb N \to \Bbb N$ be a nondecreasing function that satisfies $f(8) \geq 1$ and $f(n)\geq 2f(\lceil \frac n2-n^{2/3} \rceil)$. Can we deduce that there exists some positive constant $c$ such ...
3
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235 views

Summation convention, partial derivatives.

I've started intro to tensor calculus and I'm pracicing Einstein's summation convention. My taks is to derive the expression for third-order partial derivatives. There is a function $f(u) = F(a(u))$ ...
3
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38 views

Function estimation for a sum

I'm having a bit of a problem with finding the function that represents the sum : $$\sum_{n = 1}^\infty\frac{(-1)^{n+1}x^{n+1}}{n(n+1)}$$ I decided to differentiate it a few times and see if I get ...
3
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71 views

Simpler formula for $\frac{(pn)!}{(p!)^k(p(n-k))!}$

Let $n,p,k$ be non-negative integers ($p$ prime, but not important here). I wondered if there is a more closed-form (i.e., simpler) formula for $$A(p,n,k)=\frac{(pn)!}{(p!)^k(p(n-k))!}$$ and more ...
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0answers
190 views

Sum of prime number of compositions of functions $\sum_{n=1}^{\infty} (-1)^{n-1} f_{P_n}(\alpha)$

I noticed that the sum of certain functions composed prime-number of times sometimes converges and I'm curious what we can say about the properties of functions that do converge and those that do not. ...
3
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56 views

A question on a limit arising in Fourier series

Let $$a_k = \frac{1}{k\pi}\sum_{j=0}^m\alpha_j\sin(kT_j)$$ $\alpha_j,T_j \in \mathbb{R}, 0<T_j<\pi,m \in \mathbb{N}$. I'd like to know the limit if it exists, $$\lim_{n\to\infty}\frac{1}{\log(...
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41 views

A discrete summation in probability theory

I have to solve this discrete sum for a statistics question. This sum for $$ \frac{1}{2^N} \sum_{d=-N,-(N-2),-(N-4)}^N \frac{\lvert d\rvert N!}{(\frac {N+d}{2})!(\frac {N-d}{2})!)}$$ To do this sum,...
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79 views

Linear Algebra- $Tv_j = \sum_{i=1}^{n}v_{i}\otimes w_{ij} $

Let $V$ and $W$ be finite dimensional vector spaces over the same field. Let $T$ be a linear transformation $T : V \rightarrow V\bigotimes W$. and $B_V = \{v_1,v_2,...,v_n\}$ is a basis of $V$ and $...
3
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0answers
47 views

If $\sum_{i=1}^{\infty} a_i< \infty$, find $b_i$ such that $\sum_{i=1}^{\infty} a_i/b_i<\infty$.

Let $\{a_i\}$ be a given sequence of positive numbers such that $\sum_{i=1}^{\infty} a_i < \infty$. Is it always possible to produce a sequence $\{b_i\}$ of positive numbers such that $\sum_{i=1}^{\...
3
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0answers
116 views

Is there a closed form for $\sum_{k} \frac{1}{\cos (k)}$?

It is known that ($N$ positive integer) $$ \frac{1}{2N} \sum_{n=0}^{N-1} \frac{1}{\left[\cos\left( \frac{(2n+1)\pi}{4N} \right)\right]^2} = N $$ see for example this MSE post A curious identity on ...
3
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82 views

Matrix product calculation of squared-polynomial coefficients

Answers given in this question descirbe how if we square an order-$m$ polynomial with coefficients $a_k$: $$ p(x) = \left( \sum_{k=0}^{m} a_k x^k \right)^2 = \sum_{k=0}^{2m} c_k x^k $$ The ...
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257 views

Calculate the probability of $x$ successes in a row in n trials?

I am trying to find the probability that a person hits a baseball into a "success zone" five times in a row within $30$ total hits. The chance of hitting the success zone is $.3571$. I am not sure ...
3
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44 views

Is there a deep connection between the urn problem and the sum of powers?

The urn problem goes as follows: How many ways are there to place $k$ indistinguishable balls into $n$ distinguishable urns? The solution to the problem being $\binom {n+k-1}{k}$ (or $\binom {n+k-1}{n-...
3
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0answers
70 views

On geometric series and the logarithmic integral

With this I'm recycling a previous deleted post that contained a serious mistake (in my calculation a geometric series was not convergent). Also was interested in a diffrent question I am asking it ...
3
votes
0answers
120 views

What does $\sum_{n=0}^\infty \sqrt[2^n]{2}-1$ converge into?

I got the following sum from a friend as a challenge: $$\sum_{n=0}^\infty \sqrt[2^n]{2}-1$$ I already know it converges to approximately $1.78183863$, and the full number is irrational doesn't have a ...
3
votes
0answers
64 views

How many tests to validate an identity?

Discrete formulas such as the Faulhaber summations can be verified by evaluating them for a finite number of values. For example $$\sum_{k=1}^nk=\frac{n(n+1)}2$$ is validated by evaluating for $n=0,1,...
3
votes
0answers
213 views

Cauchy Product of Fourier Series with itself

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an $L^1$ and $L^2$-integrable function over $[0,2\pi)$ whose Fourier coefficients are real and are given by $\hat{f}$, and in particular $\hat{f}(0) = 0$. ...
3
votes
0answers
58 views

What can be said about the sum $f(x) = \sum_{m=1}^{\infty} e^{-x\sqrt{m}}\cos(x\sqrt{m})$ for $x>0$?

I recently ran into the function $f(x)$ defined by the sum $$ f(x) = \sum_{m=1}^{\infty} e^{-x\sqrt{m}}\cos(x\sqrt{m})\, $$ where $x>0$, and I've been unable to make head or tails of it either ...
3
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0answers
161 views

Sum of Complex Numbers and Modulus Inequality

Let $z_{1}, \dots, z_{n} \in \mathbb{C}$. Then, there exists a subset $S \subset \{1,\dots,n\}$ such that: $ \left| \displaystyle\sum_{j \in S}z_{j} \right| \geq \displaystyle\frac{1}{6}\sum_{j=1}^{n}...
3
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0answers
129 views

What is asymptotics of this oscillatory double sum? (Fractal Dimension problem)

The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value. However, this problem becomes particularly annoying to deal with when trying to ...
3
votes
0answers
115 views

Finding a summation involving gcd

I am trying to evaluate the following sum: $$b\sum_{i=a}^b\frac{i}{\gcd(i,b)}$$ I have solved the problem if $a=1$ but I am clueless for the case when $a$ is not $1$. For $a=1$, I used the fact that $...
3
votes
0answers
1k views

Dividing Two Sums

I have three sets of data points, let's say...$\{1,2\}\cup\{3,4\}\cup\{5,6\}$. Now suppose that I want to sum up the first point in each, and divide it by the sum of the quotients of each pair, e.g. $...