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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Show that $n^2$ is a factor of the sum of odd powers $(>1)$ of first $n$ integers

We can easily show that $n$ is a factor of the sum of $p$-th powers of the first $n$ integers , by assuming that the sum is a general polynomial of order $p+1$, and setting $n=0$, giving a zero constant term (as the sum is the same whether counted…
Hypergeometricx
  • 22,657
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show that $\sum_{i = 1}^{2k}\frac{ (-1)^{i+1}}{i} = \sum_{i = k+1}^{2k} \frac{1}{i}$

I have a proof but it does not seem elegant. Is there a more elegant solution? Thanks. Consider $X = \sum_{i = 1}^{2k}\frac{(-1)^{i+1}}{i} = X_1 + X_2$ where $X_1 = 1 - \frac{1}{2} + ... + \frac{1}{k-1} - \frac{1}{k}$ $X_2 = \frac{1}{k+1} -…
sku
  • 2,643
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How can i find the sum?

How can i calculate this serias? $$\sum_{i=1}^n \frac{2^m}{2^i}\cdot i$$ I tried to do: $$\ 2^m\cdot\left[\frac{1}{2}+\frac{2}{2^2}+...+\frac{n}{2^n}\right]$$ And i don't know how to continue.. Thank you.
user337036
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Simplifying a sum with two indices

Suppose I have $$ S_n = \sum_{1 \leq j < k \leq n } a_j a_k $$ Goal is to simplify it. This is what I got $$ S_n = \sum_{j=1}^n \sum_{k = j+1}^n a_j a_k = \sum_{j=1}^n a_j \sum_{k=j+1}^n a_k $$ Is this a correct way to rewrite this sum? Is there a…
ILoveMath
  • 10,694
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number of $\lbrace 1,2,...,53 \rbrace$ subsets with member summation divisible by $3$

In $\lbrace 1,2,3,...,53 \rbrace$ how many subsections do we have with this condition: the summation of subset members must be divisible by $3$. for example $\lbrace 1,2 \rbrace$ & $\lbrace 1,2,3,4,5 \rbrace$ count.
Rman
  • 41
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The sum to n terms of $\cos a+n\cos2a+\frac{n(n-1)\cos3a}{2!}+...$

The question is to find out the value of $$ \cos a+n\cos2a+\frac{n(n-1)\cos3a}{2!}+...$$ upto n terms. I tried to use the series expansion of $(1+x)^n$ but couldn't manipulate it to bring it in the form required.Any hints to move ahead shall be…
Navin
  • 2,605
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Evaluate $\sum\limits_{1 \le j\le k\le n} jk $

I am trying to solve an exercise from Concrete Mathematics but I seem to be stuck on the sum $$\sum_{1 \le j\le k\le n} jk $$ How to proceed? I have tried using Iverson's bracket condition like $$ [1\le j\le k\le n] = [1\le j \le n][j\le k \le n]…
K Soe
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Double summation equation

Could somebody use a simple example to explain this double summation equation? Thank you. $$\sum_{j=1}^2 \sum_{i=1}^8 x_{i,j} = \sum_{i=1}^8 x_{i,1} + \sum_{i=1}^8 x_{i,2}$$
user697911
  • 143
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Finding the formula of the sum $\frac1{1\cdot2} + \frac1{2\cdot3}+ \frac1{3\cdot4} + \cdots + \frac1{n\cdot(n+1)}$?

I am having some trouble finding out the formula for this sum: $$\text{?} = \frac1{1\cdot2} + \frac1{2\cdot3}+ \frac1{3\cdot4} + \cdots + \frac1{n\cdot(n+1)}$$ I am not sure where to start finding the formula. I know the answer is $1/(n+1)$ but how…
Hidaw
  • 971
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Is it possible to put series in other series?

I've been working on a project for quite a long time but I found myself stuck at a step where I have to multiply elements of a series by elements of another series, which is dependent on the former one. Eventually, it should look like this: …
Estagon
  • 61
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Summation of series involving factorials.

I got this question in a maths contest archive and I am completely clueless over how to start. $$\sum_{m=0}^q(n-m){(p+m)!\over m!}= {(p+q+1)! \over q!}\left(\frac{ n}{ p+1}-\frac {q}{p+2}\right)$$ I thought of transforming $\frac{(p+m)!}{m!}$ into…
Harsh Sharma
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Why is $((n-1) \mod 9)+1$ equal to summing all digits till one digit is left?

There was a question on SO on how to, in excel, sum all digits in a number until you are left with one single digit. The correct answer, in excel format, turns out to be =1+MOD(A1-1,9) which I wrote as $((n-1) \mod 9)+1$. I tried this it out with…
sowa
  • 165
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Is there a closed-form expression for this nested sum?

Is there a closed-form expression for this nested sum? $$s(n)=\sum_{i=1}^n\;\; \sum_{j=i+1}^n \sum_{k=i+j-1}^n1$$ If yes, what is it and how can it be derived?
igor
  • 473
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3 answers

Closed form of $\sum_{k=1}^{2015}(-1)^{k(k+1)/2}k$

$$\sum_{k=1}^{2015}(-1)^\frac{k(k+1)}{2}\times k$$ How to solve this. Answer provided is $0$
mnulb
  • 3,381
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How to calculate $ \sum_{n=1}^{15}n(n!) = ? $

In a contest between me and my friend, i was able to solve all the questions till he stumped me at this one. $ \sum_{n=1}^{15}n(n!) =?$ The only thing I could think of how to pursue is $n(n+1)! = n^2(n-1)!$. Preferably give hints, so i can try it…
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