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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How to change the order of summation?

I have stumbled upon, multiple times, on cases where I need to change the order of summation (usally of finite sums). One problem I saw was simple $$ \sum_{i=1}^{\infty}\sum_{j=i}^{\infty}f(i,j)=\sum_{j=1}^{\infty}\sum_{i=1}^{j}f(i,j) $$ and I can…
Belgi
  • 23,150
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A formula for the power sums: $1^n+2^n+\dotsc +k^n=\,$?

Is there explicit formula for the expression $1^n + 2^n + \dotsc + k^n\,$? I know that for $n=1$ the explicit formula becomes $S=k(k+1)/2$ and for $n=3$ the formula becomes $S^2$. But what about general $n$? I know there is a way using the Taylor…
Badshah
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How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$?

How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$, where $\mathrm{pop}(n)$ counts the number of bits '1' in the binary representation of $n$? Is there any trick to calculate the sum? From what I already have,…
tkroman
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$\frac{1}{9} + \frac{1}{99} + \frac{1}{999} + \frac{1}{9999} + \cdots ={} $?

Sum the following: \begin{align} S &= \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + \frac{1}{9999} + \cdots\\[0.1in] &= \sum_{n=1}^{\infty} \frac{1}{{10}^n - 1} \end{align} It's fairly straightforward to show that this sum converges: \begin{align} S…
John Barber
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Finite Sum $\sum_{i=1}^n\frac i {2^i}$

I'm trying to find the sum of : $$\sum_{i=1}^n\frac i {2^i}$$ I've tried to run $i$ from $1$ to $∞$ , and found that the sum is $2$ , i.e : $$\sum_{i=1}^\infty\frac i {2^i} =2$$ since : $$(1/2 + 1/4 + 1/8 + \cdots) + (1/4 + 1/8 + 1/16 +\cdots) +…
JAN
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Rules of Double Sums

What are the (most important) rules of double sums? Below are some rules I encountered - are they all correct and complete? Offerings of clear intuition or proofs (or other additions) are most welcome to remove confusion. General case:…
Anna D.
  • 525
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Sum of 1 + 1/2 + 1/3 +.... + 1/n

How do I calculate this sum in terms of 'n'? I know this is a harmonic progression, but I can't find how to calculate the summation of it. Also, is it an expansion of any mathematical function? 1 + 1/2 + 1/3 + 1/4 +.... + 1/n
Infinity
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Prove $\sum_{n=1}^{\infty}\arctan{\frac{1}{n^2+1}}=\arctan\left(\frac{\tan\pi\sqrt{(\sqrt{2}-1)/2}}{\tanh\pi\sqrt{(\sqrt{2}+1)/2}}\right)-\dfrac\pi8$

My problem: Show that $$\sum_{n=1}^{\infty}\arctan{\frac{1}{n^2+1}}=\arctan\left(\frac{\tan\left(\pi\sqrt{\frac{\sqrt{2}-1}{2}}\right)}{\tanh\left(\pi\sqrt{\frac{\sqrt{2}+1}{2}}\right)}\right)-\frac\pi8$$ This result is nice because I know the…
math110
  • 93,304
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Expansion of the square of the sum of $N$ numbers

Do I need to cite any results to use the following equality $$\left( \sum_{n=1}^N a_n \right)^2 = \sum_{n=1}^N a_n^2 + 2 \sum_{j=1}^{N}\sum_{i=1}^{j-1} a_i a_j $$ where $a_n \in \mathbb{R}\setminus{}\left\{0\right\} $ or $a_n \in…
Halil Sen
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Sums of rising factorial powers

Doodling in wolfram, I found that $$ \sum^{k}_{n=1}1=k $$ The formula is pretty obvious, but then you get $$ \sum^{k}_{n=1}n=\frac{k(k+1)}{2} $$ That is a very well known formula, but then it gets interesting when you…
chubakueno
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Intuitive reason for why $\left(\displaystyle\sum_{i=0}^n i\right)^2 = \displaystyle\sum_{i=0}^n i^3$

Is there any intuitive reason or deeper meaning to why the following equality holds? $$\left(\sum_{i=0}^n i\right)^2 = \sum_{i=0}^n i^3$$ I'm not looking for a proof of this, I'm looking for some explanation for this equality if it isn't just a…
Ziad Fakhoury
  • 2,593
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8 answers

I found this odd relationship, $x^2 = \sum_\limits{k = 0}^{x-1} (2k + 1)$.

I stumbled across this relationship while I was messing around. What's the proof, and how do I understand it intuitively? It doesn't really make sense to me that the sum of odd numbers up to $2x + 1$ should equal $x^2$.
undo_all
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4 answers

Why is this sum zero?

I have been looking at the following sum (for any positive integer $n$) $$\left(1-\frac{1^2}{n}\right) + \left(1-\frac{1}{n}\right)\left(1-\frac{2^2}{n}\right) + \left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)\left(1-\frac{3^2}{n}\right) +…
user66307
12
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2 answers

Summation by parts of $\sum_{k=0}^{n}k^{2}2^{k}$

I want to evaluate this sum $$\sum_{k=0}^{n}k^{2}2^{k}$$ by summation by parts (two times) and I need to know, if my approach was right. I know the formula for summation by parts is $$\sum u\Delta v=uv-\sum\left(Ev\right)\Delta u$$ where…
Aufwind
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12
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3 answers

Summation Skip Notation

The following notation means to sum 1 to N: $$\sum_{n=1}^N n$$ Is there a notation to not increment by one for each step, but, say, 10?
knpwrs
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