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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Summation multiplication

What is wrong with writing: $\displaystyle \Sigma_{n}\ a_n .$ $\Sigma_{n}\ b_n$ ? I understand that it does not matter what dummy variable you sum over, but I don't understand why this is seen as ambiguous
usainlightning
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When is a sum given in closed form?

Let $a>0$ be a real number. Consider a sum: \begin{equation} S_n(a) := \sum\limits_{k=0}^{n-1} \binom{k-1/3}{-1/3} \binom{k-1/3+a}{-1/3} \end{equation} Note that if $a = 1/3 + h$ where $h$ is a positive integer the term in the sum can be written as…
Przemo
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The sequence $x_1,x_2,x_3,\cdots$ is defined by $x_1=2$ and $x_{k+1}=x_k^2-x_k+1$ for all $k \ge 1$. Find $\sum_{k=1}^\infty \cfrac{1}{x_k} $

The sequence $x_1,x_2,x_3,\cdots$ is defined by $x_1=2$ and $x_{k+1}=x_k^2-x_k+1$ for all $k \ge 1$. Find $\sum_{k=1}^\infty \cfrac{1}{x_k} $ By experimenting ,I was able to prove by induction that $$\sum_{k=1}^j \cfrac{1}{x_k}=\cfrac{x_{j+1}…
Mr. Y
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Are these 3 summation expressions equivalent?

$$\sum_{i=1}^{n}f(i)+k = \sum_{i=1}^{n}\{f(i)+k\} = k+\sum_{i=1}^{n}f(i) $$ I'm more confused about the working of expression: $$\sum_{i=1}^{n}\{f(i)+k\}$$ Are all the 3 expressions equivalent?
Siddharth Thevaril
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How to simplify this summation(s)

How can I simplfiy this summation? $$ \sum_{i = 0}^{n}(a_i * \sum_{j = i}^{n} a_j) + \sum_{k = 0}^{n} a_k $$ with $a \in \mathbb{R}$
Entimon
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Evaluate an infinite sum

I've been trying to find a way to evaluate a sum and i can't. I lost some classes and now find it difficult to understand, the notes that i've been given are not specific and i've been googling for some time and can't find anything that helps. The…
Notorious Nick
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Help in explaining solution to sequence problem (finite sum)

The problem is as follows: A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are,…
mathflair
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Make explicit formula of a combined sum

I need to get the sum of the next series: $\sum_n^m(n*(\sum_n^mn)) \Rightarrow [n,m]\in \Bbb{N}$ So i am not able to make it explicit. But i can't combine the $\sum_n^m = \frac{n(n+m)}{2}$ formula inside of the sum. I need to get an explicit…
Kovács Gergely
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How find $\sum \limits _{k=1}^n \frac{1}{(k+1) \sqrt{k} + k \sqrt{k+1} }$

How find sum $\sum \limits _{k=1}^n \frac{1}{(k+1) \sqrt{k} + k \sqrt{k+1} }$ ? Maybe there is a simple way.
piteer
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Binomial Coefficients within partial sums

I need to be able to show that: $\sum_{k=i}^{n} {n \choose k} (1-t)^{n-k} t^{k-i} {k \choose i} (1-\tau)^{k-i}$ is equivalent to ${n \choose i} (1-\tau t)^{n-i}$. However I have no idea how to expand this sum out.
CasulOrnstein
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Does this summation identity hold?

Suppose we have $\sum_{n=-\infty}^{-N} f(n)$. Is this sum equal to $\sum_{n=N}^{\infty} f(-n)$? The reason why I ask this question is because I am trying to write $\sum_{- \infty}^{\infty} \frac{(-1)^n}{(2n-1)^3}$ as a summation of form…
user203867
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Does $\sum_{i \ne j}{P(\omega_i)P(\omega_j)}$ include permutations?

I recently came across the following summation in the definition of the Gini impurity in a machine learning lecture on decision trees: $$\sum_{i \ne j}{P(\omega_i)P(\omega_j)} \space\space\space i,j \in c$$ Based on the formulas used in the lecture,…
TheSchwa
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Solve the equation $1-\tan x + \tan^2 x - \tan^3 x + ... = \frac{\tan 2x}{1+\tan2x}$

How to solve this? Any advice? $$1-\tan x + \tan^2 x - \tan^3 x + ... = \frac{\tan 2x}{1+\tan2x}$$ Next step I do this $\sum\limits_{n=0}^\mathbb{\infty}(-1)^n \tan^nx = \frac{\tan 2x}{1+\tan2x} $ But I don't know next step. I am culeless, thanks…
DavidM
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Compute $\sum_{n=1}^\infty \cos(\frac{n\pi x}{L})\cos(\frac{n\pi x'}{L})$

Let $x$ and $x'$ be in the interval $[0,L]$ Compute: $$ \sum_{n=1}^\infty \cos\left(\frac{n\pi x}{L}\right)\cos\left(\frac{n\pi x'}{L}\right) $$ I remember a lot of time i have done this and if I'm not bat the result is $\delta(x-x')$ I have proved…
Juan Camilo Acosta Arango
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Solving for a variable in a summation problem?

$$x=\sum_{N=1}^TA^N$$Say I have a problem like this , how would go about rewriting this equation so that I can solve for t using x and a? I don't know what the individual variables are called so I couldn't look it up using google. I am guessing that…
NoahGav
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