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 express sum of even numbers in sigma notation

Given a positive integer number n, how can I express the sum of all positive even numbers up to n in sigma notation?
Chin
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Expectation identity interchange of infinite sums

I was deriving the identity $$EX = \sum_{n=1}^{\infty} p(X\geq n).$$ I had to use an interchange of limit sums from $\sum_{n=1}^{\infty} \sum_{i=n}^{\infty} p(X=i)$ to $\sum_{i=1}^{\infty} \sum_{n=1}^{i} p(X=i)$ but I don't know how to justify this.…
Kashif
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$n^2 + (n-1)^2 + \dots +2^2 + 1^2$ equals what?

$$n^2 + (n-1)^2 + \dots +2^2 + 1^2 = \text{??}$$ I am reading that this equals $$\frac{1}{3} n \left( n + \frac{1}{2} \right) (n+1) $$ But have no clue how.. The thing that strikes me most is the fact that the latter has a cubed factor when…
onimoni
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Sum of $\sum_{n=1}^{a}\sin\left(\left(2n+1\right)x\right)$

Why does $$\lim_{a\to\infty}\sum_{n=1}^{a}\sin\left(\left(2n+1\right)x\right)$$ seem to "fill in" the area between the curves $-\sin(x)$ and $\cot(x)\cos(x)$? The picture shows what I mean, with the green curve being the summation up to around 26…
girobuz
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Show that $\sum_{m=1}^{n-1} m(n-m) 2^{m-1}=(n-3)2^n + n + 3$ without using induction.

It is easy to show that $$\sum_{m=1}^{n-1} m(n-m) 2^{m-1}=(n-3)2^n + n+3 \tag{1}$$ using induction if one knows that $\sum_{m=1}^n m 2^{m-1} = (n-1)2^n + 1.$ But I was only able to guess the right closed-form after much trial-and-error. Is there a…
Ojas
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what is $\sum_{k=1}^{\infty} e^{\frac{1}{k}}-1-\frac{1}{k}$?

I tried to use $e^{x}=\sum_{i=1}^{\infty} \frac{x^{i}}{i!}$ and then change the order of summation but it didn't work out. I also didn't manage to prove that it converges, so I would be really grateful if someone could sort that out for me
Michael Morad
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Simple Summation Question

I don't know why I'm confused this elementary stuff, but is the following correct? $v_{1}v_{1}^{*} + v_{1}v_{2}^{*}... + v_{1}v_{n}^{*} + v_{2}v_{1}^{*} + v_{2}v_{2}^{*} ... + v_{2}v_{n}^{*} + v_{n}v_{1}^{*} + v_{n}v_{2}^{*} + ... + v_{n}v_{n}^{*} =…
Lory
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Closed form for an infinite sum

Let $\psi_q(z)$ be the q-DiGamma function defined for a real variable $\Re(z)>0$ as $$\psi_q(z)=\frac{1}{\Gamma_q(z)}\frac{\partial}{\partial z} (\Gamma_q(z))$$ where $\Gamma_q(z)$ is the q-Gamma function defined as…
Max
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expand and group like terms $(a_1+a_2+...+a_m)^n$

I know this is the multinomial theorem which is $$(a_1 + a_2 +a_3+... +a_n)^m=\sum_{k_1+k_2+...+k_n=m}\frac{m!}{k_1!\cdot...\cdot k_n!}a_1^{k_1}\cdot...\cdot a_n^{k_n}$$, but want to ask if my thought process is correct and if I made any mistakes…
John Doe
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How to show $\sum_{n=0}^\infty (-1)^n\, \frac{\Gamma \left(\frac{n}{2}+1\right)}{n! \,\Gamma \left(2-\frac{n}{2}\right)}=\frac{3-\sqrt{5}}{2}$

The following sum was used in an unrelated answer on math.stackexchange.com. $$\sum_{n=0}^\infty (-1)^n \frac{\Gamma \left(\frac{n}{2}+1\right)}{n! \Gamma \left(2-\frac{n}{2}\right)}=\frac{3-\sqrt{5}}{2}$$ How can you show this to be true?
user35671
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Infinite summation with multiple terms in denominator and variables in exponents

I need to find $\sum\limits_{n=1}^{\infty} \frac{6^n}{2^{1 + 2 n} + 3^{1 + 2 n} - 5\cdot 6^n}$, but I don't know how to do infinite summation with multiple terms in the denomiator and variables in exponents. Can anybody give me a hint? Thanks :)
CoderCookie10
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Infinite Summation $\sum_1^{\infty}\frac{1}{7^n}(\frac{1}{n^2+n})$

This came up while integrating: \begin{align}\int_0^1\frac{dx}{7^{[\frac{1}{x}]}}\end{align} after transformation $\frac{1}{x}=u$, it…
Shimura Variety
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parametrization of the result of $\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}\sum_{k=j+1}^N$ as a function of $N$

As stated in the title, what's the result as a function of N for $$\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}\sum_{k=j+1}^N 1$$
Gaston
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If you're watching a live stream and n number of minutes behind the live feed, how much time would it take while watching at s speed to catch up?

Sometimes I'll tune into YouTube livestreams a few minutes late, but I want to watch the whole thing and catch up with the live feed, so I'll start from the beginning at 2x speed or 1x speed. I'd like to find a way to calculate exactly how much time…
Tessa Painter
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Value of this infinite sum

I want to compute the value of $$\sum_{n=1}^{\infty} \frac{1}{((2n)^2 - 1)^2}.$$ I have tried evaluating the first couple partial sums, but can't find any recurrence. I don't have too many tools in my toolbox to proceed... Can anyone see how I could…
Pascal Engeler
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