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.

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How to calculate the summation of $w_i$, where $w_i = {n(n-1)\over (n-i)(i+1)}+{i-1\over i+1}w_{i-1}$

I am having trouble to calculate $$\sum_{i=1}^{n-1} w_i$$ where n is constant and $$w_i = {n(n-1)\over (n-i)(i+1)}+{i-1\over i+1}w_{i-1}$$ I am given the hint that $$ \sum_{i=1}^{n-1} w_i = n(n-1)\sum_{i=1}^{n-1} {1 \over (n-i)(i+1)}(1+…
THD
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How to solve this summation derived from an algorithm?

I have the following expression: $$\sum_{i=1}^{n}\sum_{j=1}^{i^2}\sum_{k=1}^{n/2}c=$$ Can someone explain in detail how can I solve this one?
J. Doe
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If $f(r)$ be the integer closest to $\sqrt[4]{r}$, then calculate the value of $\sum^{1995}_{r=1} \frac{1}{f(r)}$

If $f(r)$ be the integer closest to $\sqrt[4]{r}$, then what is the value of $\displaystyle\sum^{1995}_{r=1} \frac{1}{f(r)}$ ? I am thinking like this way , if we have a number $>1.5$, then its closest integer is $2,$ but I did not understand how…
DXT
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Finding the sum of $\sum_{i=0}^{2n} (-2)^i$

I am aware that the series will have a sequence of $1, -2, 4, -8, 16, \ldots$ which is the number before it, added to the next multiple of $3$. The difference of $1$ and $-2$ being $3$ and the difference of $-2$ and $4$ being $6$.
L. Li
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How to re-express sigma notation with sub index

How do you solve the expression $$\sum_{j=0}^nj\sum_{1\le i_1
useranonis
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Suppose $p(n)>0$, prove that $X$ is a linear space if and only if $\sup_n p(n)<\infty$.

$X=\{ (x_k)_{k=1}^{\infty}:\sum_1^{\infty}\mid x_k \mid^{p(n)} \}$ I am able to show that, if $\sup_n p(n)=N<\infty$, then for a scalar $\alpha$, we have $\sum_1^{\infty}\mid \alpha x_k \mid^{p(n)}\leq \mid \alpha \mid^N\sum_1^{\infty}\mid x_k…
user385459
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What is the sum of reciprocals of first $n$ Euclid numbers

We know about Euclid Number . I want to know the sum of reciprocals of 1st $n$ Euclid Number ? In that book I have been told to find the value of following series : $$ \dfrac{1}{e_1}+\dfrac{1}{e_2}+\cdots+\dfrac{1}{e_n} = ? $$ In fact, I can not…
Way to infinity
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Summation. What does is evaluate to?

What is $\sum_{n=1}^{\infty} \frac{a_{n}}{4^{n+1}}$ if $a_{n+2}=a_{n+1}+a_{n}$ and $a_{1}=a_{2}=1$?
scubnoob
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square of sum inside summation notation

I am looking to find out how the derivation went in this computation $$ \frac{1}{n-1} \sum (x_i - \bar{x})^2 = \frac{1}{n-1} \left( \sum x_i^2 - n\bar{x}^2 \right) $$ The exercise belongs to sample distribution section but that's not what bothers…
johndoe123
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Determine if sum $\sum_{k=1}^\infty\frac{k!b^k}{(a+kb)\cdots (a+1\cdot b)}$ diverge

This is a special case, so we assume that $a>b$ and I need to figure out if the sum $\sum_{k=1}^\infty\frac{k!b^k}{(a+kb)\cdots (a+1\cdot b)}$ diverge to infinity or not. I have tried a lot of things without succes so a hint would be appreciated.
Hamid Mohammad
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How to sum a series with different index

I am new to this site so not sure if this type of question is appropriate. I know that the sum of a geometric series can be written like this: $$ S = \sum ^n_{k=1} a^k = \frac{a^1 - a^n}{1-a} $$ How does this change however if the power of each term…
user513534
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Simplifying $ \sum _{k=-n}^n k^2 q^k $

How can we simplify $ \sum _{k=-n}^n k^2 q^k $? Is there any nice expression?
Asghar
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Closed form for $\sum_{j = 0}^k \frac{j}{(k-j)!}$

Is there a closed form for $$\sum_{j = 0}^k \frac{j}{(k-j)!}$$ for $k \in \mathbb{N}$.
user317721
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Expressing $\sum_{n=1}^\infty\sum_{r=1}^n \frac 1{(2n+1)(2n+2)r}$ as two summations

The question here requires the evaluation of the following summation: $$\sum_{n=1}^\infty\frac 1{(2n+1)(2n+2)}\sum_{r=1}^n \frac 1r\tag{1}$$ The answers provided (and also wolframalpha) show that the solution is $$\frac {\pi^2}{12}-(\ln…
Hypergeometricx
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