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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Closed form to series

Is there a closed form of the series $$ \sum_{d=0}^D F^d $$ where D is a finite integer, not $\infty$, like in power series?
Make42
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Calculating the summation $\sum_{n=0}^{\infty} \frac{n^3}{3^n}$

Evaluate $$\sum_{n=0}^{\infty} \frac{n^3}{3^n}.$$ I started my using casework, listing out some terms when plugging values in. $0, \frac{1}{3}, \frac{8}{9}, 1.$ I noticed that there is no common ratio involved in these terms. I'm not sure if I can…
Frost Bite
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Truncated sum of $e^x$

I know that: $$\sum_{i=0}^\infty \frac{(x\ln{2})^i}{i!}=e^{x\ln{2}}=2^x $$ from the Taylor series for $e^x$. How can I find an approximation or asymptotic expression for the terminated sum: $$\sum_{i=0}^{x-1} \frac{(x\ln{2})^i}{i!}$$ I believe that…
Haran
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$\sum \limits_{k~ \in~ K_e} \frac{a^k}{k!} - \sum \limits_{k~ \in~ K_o} \frac{a^k}{k!} = \sum \limits_{k=0}^{\infty} \frac{(-a)^k}{k!}$

Given: $$K_e = \{\text{even numbers: 0, 2, 4, ...}\}$$ $$K_o = \{\text{odd numbers: 1, 3, 5, ...}\}$$ How to prove this equality is true? $$\sum \limits_{k~ \in~ K_e} \frac{a^k}{k!} - \sum \limits_{k~ \in~ K_o} \frac{a^k}{k!} = \sum…
pico
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Identity of sum $\sum_{a}^{b}x=-\sum_{b}^{a}x$

Does $\sum_{a}^{b}x=-\sum_{b}^{a}x $ is correct?
violettagold
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How to solve this sum of fractions in terms of $k$?

I'm interested in solving the series $$\frac{2}{1\cdot3}+\frac{3}{3\cdot5}+\frac{4}{5\cdot7}\cdots+\frac{n+1}{(2n-1)(2n+1)}.$$ That is, $$\sum^n_{k=1}\frac{k+1}{(2k-1)(2k+1)}.$$ Can we get a concise representation for this summation?
Andeanlll
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Compute the value of 2 summation equations

Questions: For a, I got Is my approach correct? As for b, how do I compute it?
user741540
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Any hope for evaluating, or accurately approximating, this sum for large $n$?

The sum $\sum_{a,b,c,d,e,f=1\wedge !(a=b \wedge c=d \wedge e=f)}^{n} (e-f)^2/((a-b)^2+(c-d)^2+(e-f)^2)^{3/2}$ was presented with a desire to evaluate it for $n$ up to 100. Clearly doing this directly becomes preposterous for large $n$, with the…
rasher
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Is there any way to calculate the following sum:$G(n)=\sum_{i=0}^ni\cdot f(i)$?

Suppose we have some formula $f(x)$ and we are able to work out the sum of first $F(n)=\sum_{i=0}^nf(i)$ with some nice formula $F(n)$; Now suppose we want to work out tha following sum $G(n)=\sum_{i=0}^ni*f(i)$ Is there any way to work out / derive…
blindProgrammer
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Basic summation to formula, trying to figure out how formula was derived

This comes from a coding problem on leetcode where Big O notation is being calculated. I'm trying to understand how the left side becomes the right side here: $\displaystyle\sum_{i=0}^{n-1} {\sum_{j=i+1}^n} (j-i) = \sum_{i=0}^{n-1}…
blindsnowmobile
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How to simplify the expression below

I was stuck with simplifying this expression: $$\sum_{i=1}^{2^{n}} \log (i) $$ I was told that it equals to $$n\times2^{n}$$ Thank you for your help!
FAF
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Find the sum of $\sum_{k=1}^n\left(\frac{1}{(k+1)^2}-\frac{1}{k^2}\right)$

Hi how do I fin the sum of $$\sum_{k=1}^n\left(\frac{1}{(k+1)^2}-\dfrac1{k^2}\right)$$ I tried everything and I can't get around it..
Hai
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Evaluate $3^x$ from $0$ to $ n$. $\sum_{i=0}^{\log_{3}(n)} 3^i $

I really don't even know where to start, is there some way to interchange the upper limit and the term to sum over?
DennisB
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Sum of three means

Apologies im sure this is a very simple math question ... How do I represent the sum of three means in mathematically notation. $p =$ Mean$(x_1,x_2,x_3)$ + Mean$(y_1,y_2,y_3)$ + Mean$(z_1,z_2,z_3)$ Here is my best effort $\displaystyle\sum_{i=1}^{3}…
Bat
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Summation Reduction

Does $\sum\frac{y_i}{n}-\sum(\frac{x_i-\bar x}{(x_i-\bar x)^2}\cdot y_i)\sum\frac{x_i}{n}$ reduce to: $\sum[y_i(\frac{1}{n}-\frac{x_i-\bar x}{(x_i-\bar x)^2}\sum\frac{x_i}{n})]$ Basically, what does $\sum a-\sum b\sum c$ reduce to? edit: it should…
cap
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