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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Can anyone help me with this finite sum?

I have to calculate the sum $\displaystyle\sum_{k=1}^n \displaystyle\frac{3^k}{3^{2k+1}-3^k-3^{k+1}+1}$ We can re-write the sum as follows $\displaystyle\sum_{k=1}^n \displaystyle\frac{3^k-1+1}{(3^{k+1}-1)(3^k-1)}$ And then we…
Juan David Samboní
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Find the properties of the sum $\sum_{k=0}^n (-1)^k\binom{m+1}{k}\binom{m+n-k}{n-k}$

I have to show that $$\displaystyle\sum\limits_{k=0}^n (-1)^k\binom{m+1}{k}\binom{m+n-k}{n-k} = \begin{cases} 1\ \text{if}\ n=0 \\ 0\ \text{if}\ n>0 \end{cases}$$ My try: I have tried to use snake oil method $$\sum_{k=0}^n…
Sayan Chattopadhyay
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Simplified form for a Newton's sum? $a_1^k + a_2 ^k + \ldots + a_n^k = k $ for $k=1,2,\ldots,n$.

Let $ a_1, a_2, a_3, \ldots , a_n $ be complex number satisfying $ \displaystyle \sum_{j=1}^n a_j ^k= k $ where $ k =1,2,\ldots, n $. Prove (or disprove) that $\displaystyle \sum_{j=1}^n a_j ^{n+1} $ can be expressed $ \dfrac{K}{n!} $, where $K$ is…
GohP.iHan
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Least degree polynomial and Newton's sum? $a_1^k + a_2 ^k + \ldots + a_n^k = k $ for $k=2,3,\ldots,n+1$.

Let $ a_1, a_2, a_3, \ldots , a_n $ be complex number satisfying $ \displaystyle \sum_{j=1}^n a_j ^k= k $ where $ k =2,3,\ldots, n+1 $. Prove (or disprove) that the least degree polynomial with integer coefficients that has a root $\displaystyle…
GohP.iHan
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Need to understand this summation with max notation

Firstly, apologies needed for my math description if it does not sound right. I have come across a paper where I saw a summation notation with a max function in it which I am little confused to understand. The formula is as…
tushersuvro
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Pull a term out of a double sum

I'm trying to rewrite a double sum in the following format $$ \sum_{l=0}^\infty \sum_{n=0}^\infty z^{n-2l} g(n,l) = \sum_{m=-\infty}^\infty z^m h(n) $$ For some $h(n)$, which will probably involve a second sum. In other words, I'm trying to pull the…
smörkex
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Sum of $1/n^k$ of the first $\log P$ numbers

In a Udacity course I'm told the following: $\sum_{i=1}^{\log_2 (P)} 1/2^i = (P-1) /P $ I've checked that it's true by entering it into Wolfram Alpha: https://www.wolframalpha.com/input/?i=sum+1%2F2^n,+n%3D1+to+log%282,P%29 Could someone help me…
Avatar33
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Why is $\sum_{i=0}^{+\infty} a^{i}i=\frac{a}{(1-a)^{2}}$?

I saw this series in some mathematical proofs but I couldn't find why $\sum_{i=0}^{+\infty} a^{i}i=\frac{a}{(1-a)^{2}}$
sajjad
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Finding The maximum and minimum value of a summation given a condition

If $a_i = 1 - \frac{1}{N_i}$ and $\sum\limits_{I=0}^k{N_i} = n$, then what is the maximum and minimum values of $\sum\limits_{I=0}^k{a_i}$? Please help, I've tried to solve it but then I got confused. I think I may of found the minimum value to be…
Harry Obey
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Combinatory sum of multiplications

Suppose i have $N$ variable. In a sum, i have terms each consist of combination of n variable. Each variable(they appear only once in one term) is to be multiplied to get term. How can i write the sum in a compact way (in terms of sigma maybe)?…
Salihcyilmaz
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How to calculate $\sum^{n-1}_{i=0}(n-i)$?

How to calculate $\sum^{n-1}_{i=0}(n-i)$? $\sum^{n-1}_{i=0}(n-i)=n-\sum^{n-1}_{i=0}i=n-\sum^{n}_{i=1}(i-1)=2n-\frac{n(n+1)}{2}$ I am sure my steps are wrong. Could someone show me how to correct the procedure?
CoolKid
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Double Sigma with nested index, $i$

I am trying to solve this double sigma but my answer doesn't seem right. $$\sum\limits_{i=1}^n \sum\limits_{j=i}^{n^2}1=-i\sum\limits_{i=1}^n n^2 = -in^3$$
ocram
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Summation of stopping point

I'm learning summation and I need help with the following sum: $\sum\limits_{i=0}^{n-2} n$ What I thoughts is, since $n$ don't change, my sum will be $S_n$ = $n_1$ + $n_2$ + $n_3$ + ... + $n_{n-2}$ Then: $2S_n$ = ($n_1$+$n_{n-2}$)$n$ $2S_n$ =…
G. Castro
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Find the closed form for the double sum $\sum_{i=1}^n \sum_{j=i}^n 2j$

This is what i get: $$n^3 + n^2 - n(n+1)(2n+1)/6 + n(n+1)/2$$ When I simplify : I get : $(1/3)n(2n^2+3n+1)$ Is anyone else getting the same result.
jay6601
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summation related to reciprocals

I know that $\sum\frac{1}{n^2}=\frac{\pi^2}{6}$ in a test this was given and we were asked to find $\sum \frac{1}{(2n+ 1)^2}$ starting with $n=0...\infty$. Now i am grade $11$ student and have been here for a while so i know its proved by Euler if…
Archis Welankar
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