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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Limit of $\sum\limits_{i=1}^n\sin\left(\frac{i}{n^2}\right)$

Compute $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\sin\left(\frac{i}{n^2}\right)$$ Using Taylor expansion for $\sin x$, I know that this is…
lightfish
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Formula for summation of integer division series

Consider '\' to be the integer division operator, i.e., $a$ \ $b = \lfloor a / b\rfloor$ Is there a formula to compute the following summation: N\1 + N\2 + N\3 + ... + N\N
AncientEgypt
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Proving $\left(\sum_{n=-\infty}^{\infty}x^{n^{2}}\right)^{2}=1+4\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{x^{-\left(2n+1\right)}-1}$

$$\left(\sum_{n=-\infty}^{\infty}x^{n^{2}}\right)^{2}=1+4\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{x^{-\left(2n+1\right)}-1}$$ Proving: $$\left(1+2\sum_{n=1}^{\infty}x^{n^{2}}\right)^{2}=1+4\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n+1}}{1-…
Miracle Invoker
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Simplifying $\displaystyle f(n)=\frac{\sum^{n}_{k=0}\sin(\frac{k+1}{n+2}\pi)\sin(\frac{k+2}{n+2}\pi)}{\sum^{n}_{k=0}\sin^2(\frac{k+1}{n+2}\pi)}$

For a non negative integer $n,$ let $$f(n)=\frac{\sum^{n}_{k=0}\sin\left(\frac{k+1}{n+2}\pi\right)\sin\left(\frac{k+2}{n+2}\pi\right)}{\sum^{n}_{k=0}\sin^2\left(\frac{k+1}{n+2}\pi\right)}$$ Assuming $\cos^{-1}(x)$ takes values in $ [0,\pi]$. Which…
Priti Bisht
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Two double sums must be equal, struggle with indices

Empirically, these $2$ sums are supposed to be equal: $$\sum_{k=0}^n \sum_{j=0}^k a_j b_{k-j} = \sum_{k=0}^n a_k \sum_{j=0}^{n-k} b_j$$ I failed to prove it with a "change of index" and now I am thinking the only possible way to solve this is to…
niobium
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Evaluating $ \sum_{a=1}^{\infty}\sum_{b=1}^{\infty} \frac{a^2b^2}{\sinh(\pi(a+b))}(-1)^{a+b} $

First of all, I would like to thank you for the consultation and the opportunity to ask for help on very serious issues. I'll go straight to the question itself. Task: It is necessary to calculate the sum of the following…
Alex
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Is there a closed form solution for partial sums of $1/(2^{2^0}) + 1/(2^{2^1}) + 1/(2^{2^2}) + \ldots$

Title says it all, this is such a classical looking series, $$\frac1{2^{2^0}} + \frac1{2^{2^1}} + \frac1{2^{2^2}} + \ldots.$$ So, I was just wondering, is there a closed form solution known for the partial sums? If so, can someone post? I've…
user2566092
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Are there any rules relating the sum of a function, and the sum of the inverse of that function?

I'm in university and I've hit summation notation for the first time. I haven't studied series before due to a combination of moving universities (they each thought the other had taught this to me), and when I was in high school this wasn't part of…
Mcbestington
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What is the closed form of this summation?

$$ f(x) = \sum_{n\ge 0}\binom{n+k}{2k}x^n $$ $k$ is a constant integer. First I want to find the relation between $\binom{n+k}{2k}$'s, but failed, then I change $f(x)$ to $f(k,x)$, want to figure the relations of $f(k,x)$, but still cannot find it.…
delta
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Interchanging infinite and finite sum, positive terms

My textbook claims that on the extended reals and for a nonnegative infinite double sequence $(a_{n, m})$ we have \begin{align*} \sum_{n = 1}^{\infty} \sum_{m = 1}^{n} a_{n, m} = \sum_{m = 1}^{\infty} \sum_{n = m}^{\infty} a_{n, m} .\end{align*} As…
Richard
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Summation starting from 0

$$ \sum_{i=0}^{n}(i) $$ This seems pretty basic, but I'm starting with the subject and the only formula I have to use for these kind of problems starts the summation at 1, like this. $ \sum_{i=1}^{n}(i) $ = $\frac{n(n+1)}{2}$ Is the same formulate…
NearFinished_CS
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Determine how long a sum of $\frac{1}{n}$ is needed to reach a limit

Since $$ \lim_{n\to\infty} \sum_{i=1}^n {1 \over i} = \infty $$ it should be possible to find the $n$ where the sum reaches a certain number. Given $c$ determine $k$ where $$ \sum_{i=1}^k {1 \over i} < c \leq \sum_{i=1}^{k+1} {1 \over…
Ole Tange
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Finding $\sum_{r=1}^n rn^{r-1}\prod_{k=1}^r(n+k)^{-1}$

I was trying to find the answer to this question $$ S= \sum_{r=1}^{n}\frac{rn^{r-1}}{\prod_{k=1}^{r}(n+k)} $$ I tried finding a series which S is a derivative of, seeing that the numerator is power rule like, but wasn't able to do so. Solutions or…
Prajwal Muralidharan
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How to calculate alternating double sum

How do I determine the value of the following alternating sum (converges by Leibniz): $$ \sum\limits_{n = 0}^{\infty}\sum\limits_{k = 0}^{2n}{2n\choose k}(-1)^k (p)^{k+1},\quad p \in (0, 1) $$ I don't have any idea how one can tackle such a sum as I…
Jacob
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How to prove this summation?

How to prove this summation without mathematical induction? Thanks a lot! I can't figure out a way to reduce n to $\sqrt{n}$. $$ \sum_{i=1}^{n}\left\lfloor \frac{n}{i} \right\rfloor = 2\sum_{i=1}^{\left\lfloor \sqrt{n} \right\rfloor}\left\lfloor…
delta
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