Mathematics Stack Exchange
Mathematics Stack Exchange

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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How to sum $a_n$ from 1 to infinity?

Where $a_n = (1 - a_{n-1})(.1)$ when n is even, $= (1 - a_{n-1})(.2)$ when n is odd and $a_1 = .3$. Thank you.
RHS
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Multiplying using Summation rule

Is the sum $$ \sum_{i=1}^{2}\sum_{j=1}^{2}x_i $$ equals to $$\sum_{i=1}^{2}x_i ?$$
MrHumphey
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Splitting Summation

While studying for my course, I found following rule in the web (below image). My professor seems to assume everyone know this, but I don't understand why this works. It says $$1 + 2 + 3 + \cdots + n= [ 1 + 2 + 3 + \cdots + n/2 ] + [n/2 +…
Alex Kim
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Partial sum of pochhammer symbols and fractial

Is the following evaluation of the sum correct? $$\sum_{n=0}^k \frac{(\frac{1}{2})_n}{n!} \frac{(\frac{1}{2})_{k-n}}{(k-n)!} =1.$$ I don't see how this is immediate from the definitions.
User1825
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Convergence of this sum?

A very simple example from my textbook $$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+\frac{1}{8^2}+\frac{1}{9}+\frac{1}{10^2}+\cdots$$ or say $a_n=1/n^2$ if $n$ is not a perfect square, otherwise $a_n=1/n$.…
Page David
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Help with understanding summation formula breakdown

$S_{xy} = \sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y}) = \sum_{i=1}^n x_iy_i - \frac{1}{n}(\sum_{i=1}^n x_i) (\sum_{i=1}^n y_i)$ I don't understand how it becomes $\frac{1}{n}(\sum_{i=1}^n x_i) (\sum_{i=1}^n y_i)$ $S_{xx} = \sum_{i=1}^n (x_i-\bar{x})^2 =…
Bayman
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squaring a summation syntax

In the following summation: ${\sum_{i=1}^{N}\left(x_{i}-\bar{x}\right)^{2}}$ Do I first add all the $x_i - \bar{x}$ then I square the sum? Or do I square each $x_i - \bar{x}$ as I add them up? Sorry for the silly question -- but the syntax is…
Katie Melosto
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Help with understanding summation formula

Below is part of the proof to show $E[S^2] = \sigma^2$. I don't understand how $2\bar{Y} \sum_{i=1}^n {Yi} = 2n\bar{Y}^2$?
Bayman
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Change of variables for sumations.

I'd like to know a method to do a change of variables for sumations; as far as I am concerned it should be analogous to change of variables when integrating a continuous function. I got this summation: \begin{equation} \sum_{k=-\infty}^n…
CoolJedi132
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Formula for $\sum_{i=1}^n \lfloor i\sqrt{2} \rfloor $

I'm trying to find a faster way to compute this: $$\sum_{i=1}^n \lfloor i\sqrt{2} \rfloor $$ I doubt that there is a compact form, but there must be some way to calculate this faster than just iterating over i. If it wasn't for the floor, it would…
Finn
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Intuition behind splitting sums into sub sums

At 11:15 of this video, Michael Penn takes this sum, $$ \sum_{i=1}^{n} \sum_{j=1}^{i} \frac{1}{ \alpha_j \alpha_{i-j} }$$ And splits into three sums, $j=i-j$ , $ ji-j$ but I don't understand, what's the intuition behind this? Like,…
tryst with freedom
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Solving Double Summation

I came across the following summation but I don't know how to solve it: $\sum_{i=1}^n\sum_{j=i}^n (j + 1)$ I know how to do double summations, but I never saw double summations that are linked like this before though.
tteo57
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Can't understand a step in summation

5(b) $$\frac{1}{(2k-1)(2k+1)(2k+3)}= \frac{A}{2k-1}+\frac{B}{2k+1}+\frac{C}{2k+3}$$ $$1=A(2k+1)(2k+3)+B(2k-1)(2k+3)+C(2k-1)(2k+1)$$ $$\left. \begin{array}{ll} k=\frac{1}{2}\Rightarrow& 1=8A\\ k=-\frac{1}{2}\Rightarrow&…
Mouvre
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Evaluate: $\sum_{n=1}^{10}n\cdot (\sum_{r=1}^{10}\frac{r^2}{r+n})$

Evaluate: $\sum_{n=1}^{10}n\cdot (\sum_{r=1}^{10}\frac{r^2}{r+n})$ using the property of sigma operator I put $n$ in second sigma So it becomes: $\sum_{n=1}^{10}\cdot (\sum_{r=1}^{10}\frac{r^2n}{r+n})$ Which property I need to use now?
user69608
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need to find sum of $\sum_{k=1}^n \frac{(k-1)2^k}{k(k+1)}$

need to find sum of $$\sum_{k=1}^n \frac{(k-1)2^k}{k(k+1)}$$ this is much harder as I see. Or maybe it's my eyes, because I'm new in studying sums. I tried this one using differences , factorial powers, didn't help.
Delta Account
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