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Questions tagged [summation]

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

The notation $\sum\limits_{i=1}^na_i$ means $a_1+\ldots +a_n$.

Use for sums of infinite series and questions of convergence; use for questions about finite sums and simplification of expressions involving sums.

17770 questions
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Distributive Nonidentity Einstein Summation

From Schaum's Outline to Tensor Calculus Chapter 1, Example 1.8 — $a_{\large{ij}}(x_{\large{i}} + y_{\large{j}}) \neq a_{\large{ij}}x_{\large{i}} + a_{\large{ij}}y_{\large{j}}$. I was trying to grok why Einstein summation turned this into a…
user53259
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Showing that the sum $\sum_{k=1}^n \frac1{k^2}$ is bounded by a constant

How can I shows that the summation of $1/k^2$ from k=1 to n is bounded above by a constant? I could bind it by the geometric series from k=0 to n and add 1 to $(1/k^2)$ to get the ratio, r, and get $A_0 (1/(1-r)) $. So is that going to be summation…
user1766888
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Problem with Free Index in Einstein Summation Notation

From http://www.physics.ohio-state.edu/~ntg/263/handouts/tensor_intro.pdf: Rules of Einstein Summation Convention — If an index appears (exactly) twice, then it is summed over and appears only on one side of an equation. A single index (called a…
user53259
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Which of the following is the sum of an infinite geometric sequence whose terms come from the set $(1, {1\over 2},{1\over 4},.... {1\over 2^n}) $?

Which of the following is the sum of an infinite geometric sequence whose terms come from the set $(1, {1\over 2},{1\over 4},\ldots{1\over 2^n}\ldots) $? Given options are $a){1\over 5} \ \ b){1\over 7} \ \ c){1\over 9} \ \ d){1\over 11}$ But I got…
Chris
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Finding $\sum_{x∈X} x^2$ given $\sum_{y∈Y}y$ for each $Y⊆X$ such that $|Y|=\frac{|X|}{2}$?

Suppose we have a finite set of real numbers $X$, and we want to compute the sum $\sum_{x \in X} x^2$, however we don't know what any of the $x \in X$ are. Instead, assume that $|X|$ is even, and suppose for any $Y \subset X$ such that $|Y| =…
Michael Gilbert
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Question on Summation of a equation

Find $$\sum_{x= 1/2}^{25 / 2} x^3 + 3x + 2.$$ Are there any simpler methods to solve this summation ? I will be thankful to any helps. Counting is in the format of ${1 \over2}, 1, {3 \over2},.... $
user78815
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$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{m²n}{n3^m +m3^n}$

$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{m²n}{n3^m +m3^n}$. I replaced m by n,n by m and sum both which gives term $\frac{mn(m+n)}{n3^m +m3^n}$.how to do further?
user69608
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Solution to $\sum_{0\leq n_1\leq n_2\leq\cdots\leq n_\beta\leq\gamma}{(-1)^{n_1+n_2+\cdots+n_\beta}}$

I am trying to solve summation I have stated in title, $$\sum_{0\leq n_1\leq n_2\leq\cdots\leq n_\beta\leq\gamma}{(-1)^{n_1+n_2+\cdots+n_\beta}} $$ where $\beta,\gamma\in\mathbb{N}$. I have tried some methods, such as taking this summation as…
Kenny Kim
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Proof that $\sum\limits_{n=1}^\infty\frac1{n2^n}=\ln2 $?

Using approximation with a spreadsheet, I see that: $$\sum_{n=1}^\infty \frac{1}{n\ 2^n} = \ln 2$$ Is there a proof of this?
Don Del Grande
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Value of summation $ij$

What is the value of $\sum \limits_{1 \le i < j \le 10} ij$?
Zero
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Changing index in summation

This question seems simple but it's been eating me for 20 mins. on a book I seen something like this : $$ \sum_{k=8}^{\infty}\left(\frac{5}{6}\right)^{k-1}\frac{1}{6} =…
A.Y
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Rewriting a double summation.

If I am given this function: $$f(x) = \sum_{i = 1}^{\infty} \frac{1}{i^x}$$ Is there a way to rewrite: $$g(x) = \sum_{ j = 1}^{\infty} \sum_{i = j}^{\infty} \frac{1}{(i \cdot j)^x}$$ In terms of f(x). By simple arthimatic I know that $f(x)^2 = 2…
Nimish
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Find the sum of $k/2^k, k=1$ to $n$

Let $S=1/2+2/2^2+3/2^3+...+n/2^n$ I try searching on the internet and see only the version of $k=1$ to infinity. I put this equation on Wolfram Alpha and get $(2^{n+1}-n-2)/2^n$ but I dunno how to do that. Please help
user635988
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How to effectively calculate $(1/\sqrt1 + \sqrt2) + (1/\sqrt2 + \sqrt3) +\cdots + (1/\sqrt{99} + \sqrt{100})$

I have this series: $$\frac{1}{\sqrt1 + \sqrt2} +\frac{1}{\sqrt2 + \sqrt3} +\frac{1}{\sqrt3 + \sqrt4} +\cdots+\frac{1}{\sqrt{99} + \sqrt{100}} $$ My question is, what approach would you use to calculate this problem effectively?
Le Chifre
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What criterion of convergence should I use here?

I'm trying to check if the series $$\sum_{n=1}^{\infty} \sqrt[3]{n^3+n}-n$$ does converge and I don't know which criterion should I use. D'Alembert doesn't help, what's more I can't limit this series with a sense to use comparative criterion. The…
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