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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Trying to understand a double summation formula

I am trying to understand a double summation mentioned in this paper: https://doi.org/10.1016/0024-3795(95)00696-6. How can we prove that this is true? $$\sum_{l = 1}^n\;\; \sum_{k = l}^{n+l} = \sum_{k \,=\, 1}^n \;\; \sum_{l \,=\, 1}^{k} \,+\,…
Gokul
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Summation using binomial coefficient

I am trying to calculate the following using binomial coefficients and summation, but my memory is standing in the way: $$ \sum_{k=1}^n {k} * 2 ^ {k - 1} $$ Thanks! With great help I got to: $$ (n - 1)* 2 ^{n} - 1 $$ Can you please confirm this?
flavian
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What do the sum of the reciprocal of n squared make if n is a natural number?

I’ve recently found out that $\sum_{n=1}^{\infty}\frac{1}{n^2+n}$ makes 1, since it becomes $\frac{1}{2}, \frac{2}{3}$ and so on. After then, I’ve became curious if I do the same thing with the reciprocal of n squared, or $$\sum_{n=1}^{\infty}…
user812072
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When is this sum equal to $0$?

When, for arbitrary positive integers $m$ and $n$, is the following sum equal to $0$? $$ \sum_{i=0}^{mn-1} (-1)^{\lfloor i/m \rfloor +\lfloor i/n\rfloor} $$
arshajii
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Simplifying $\sum\limits_{k=1}^{n-1} (2k + \log_2(k) - 1)$

I'm trying to simplify the following summation: $$\sum_{k=1}^{n-1} (2k + \log_2(k) - 1)$$. I've basically done the following: $$\sum_{k=1}^{n-1} (2k + \log_2(k) - 1) \\ = \sum_{k=1}^{n-1} 2k + \sum_{k=1}^{n-1} \log_2(k) - \sum_{k=1}^{n-1}…
Mythio
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Step by step for solving $\sum^n_{i=1}{(a+b)^i}$?

Wolfram and a differential equations book that I read give me this form as solution $$\ \sum^n_{i=1}{(a+b)^i} = \frac{(a+b)((a+b)^n-1)}{(a+b-1)}$$ However, I would like to get there step by step, in order to understand the logic between expression…
ilibarra
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Sums over the positive integers

Can anyone see why it is that if $a$ is large, then $$\log (\sum_m\sum_n \exp(-knm/a)))$$ where $k$ is a constant and $n,m$ take values $1,2,3,...$, can be approximated by $${a\pi^2\over 6k }$$? Cheers! As Gerry suggested, it would probably help if…
user61472
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Formula for calculating the sum of a series of function results over a specific range of inputs?

Let's say I have a simple function, something like this: $f(x) = 5x + 30$ And I want to know the sum of the results of $f(x)$ for all consecutive integers $x$ from $Y$ to $Z$, where $Y$ and $Z$ are arbitrary values greater than $1$. For example,…
Mason Wheeler
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Relevance of upper vs lower sum indices

I read this related thread but it doesn't give me a satisfactory answer to the following question: Must it be true that the order of indices in a sum is relevant? A finite sum is essentially adding up all elements in a set. Who cares it that set…
avikarto
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Does changing the order of sigma notation matter?

Can you change the order of summation like this and play around? If no, then what does it change? $$ \displaystyle\sum\limits_{i=a}^{b} \sum\limits_{j=p}^{q} f(i) g(j) =\sum\limits_{j=p}^{q} \sum\limits_{i=a}^{b} f(i) g(j)=\sum\limits_{i=a}^{b} f(i)…
William
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Triangular Summation $\displaystyle\sum_{i=0}^n\sum_{j=0}^i (i+j)=3\sum_{i=0}^n\sum_{j=0}^i j$

It can be easily shown that the summation $$\sum_{i=0}^n \sum_{j=0}^i (i+j)\tag{*}$$ is equivalent to $$\frac 12 n(n+1)(n+2)$$ which can also be written as $$3\binom {n+2}3$$ This is the same result as the…
Hypergeometricx
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Summation of n(2^n)

I was doing a question in which i had to find the summation of the expression $n(2^n)$ from n=1 to n=9. I used wolfram alpha to calculate thid sum, but i was wondering if there is an easier way to calculate it?
Aaryan Dewan
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Show that $\left(\sum_{r=1}^n r\right)^2=\sum_{r=1}^n r^3$ without expanding to closed form

It is well know that $$\left(\sum_{r=1}^n r\right)^2=\sum_{r=1}^n r^3$$ i.e. $$(1+2+3+\cdots+n)(1+2+3+\cdots+n)=1^3+2^3+3^3+\cdots+n^3$$ and this is usually proven by showing that the closed form for the sum of cubes is $\frac 14 n^2(n+1)^2$ which…
Hypergeometricx
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Finding closed form of $ S(n)= \sum_{i=0}^n a^{q^i} $

I'm trying to find a closed form of $ S(n)= \sum_{i=0}^n a^{q^i} $ for positive constant params $a, q, n$. I searched it in google but I couldn't find any. I searched it in wolframalpha, but I didn't get any result.I thought there is some set like…
user2377766
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Partial Sum of Negative Binomial Coefficients multiplied by Rising Power of $-\frac 12$

How can it be shown that : $$\sum_{r=0}^n\binom {-n-1}r\left(-\frac 12\right)^r=\sum_{r=n+ 1}^\infty\binom {-n-1}r\left(-\frac 12\right)^r=2^n$$ ? This is confirmed by wolframalpha here and here. We know that $$\sum_{r=0}^\infty\binom…
Hypergeometricx
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