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 solve this summation? from i+1 to n

Can you explain me how to solve this summation? $$ \sum_{j=i+1}^{n}j $$ I don't know which property to apply, I thought that maybe this one: $$ \sum_{i=1}^{n}i=$$ $$ \frac{n(n+1))}{2}$$ But it goes from 1 to n, not i+1 to n
Kevin Ramirez Zavalza
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$\sum_{k=1}^{∞} k(1-p)^{k-1}+\sum_{k=1}^{∞} kp^{k-1}$ for $0 < p < 1$

$$\sum_{k=1}^{∞} k(1-p)^{k-1}+\sum_{k=1}^{∞} kp^{k-1} $$ where $0
JoZ
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Proving formula for repeated geometric sum

I'm attempting to compute a repeated geometric sum $$ \sum_{k_0=0}^m \alpha^{-k_0} \sum_{k_1=0}^{k_0} \alpha^{-k_1} \sum_{k_2=0}^{k_1} \alpha^{-k_2} \dots \sum_{k_q = 0}^{k_{q-1}} \alpha^{-k_q}. $$ If you start to explicitly compute the sums from…
Afsa
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How do solve an sum of numbers to a common power?

How would solve for $a$ in this equation without using an approximation ? is it possible? where $x>0$ and $0
Preshen Moodley
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Big theta of summation

I had a quick question with which I am not sure how to get started. I've done some time complexity stuff that involves big-oh, omega and theta notation but nothing with sums. Here is the question: $$ f(n) = \sum_{k=1}^n {k^{3}\over2^{k}} $$ Prove…
Jo Dane
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Finding closed form for $\sum_{k=1}^n k2^{k-1}$

I am trying to use the perturbation method to find a closed form for: $$ S_n = \sum_{k=1}^n k2^{k-1} $$ This is what I’ve tried so far: $$ S_n + (n+1)2^n = 1 + \sum_{k=2}^{n+1} k2^{k-1} $$ $$ S_n + (n+1)2^n = 1 + \sum_{k=1}^{n} (k+1)2^{k} $$ $$ S_n…
Diego Marques
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Simplification of some summations

Simplify $$\sum_{n=a+1}^{2a}A_n- \sum_{n=2}^{a}A_n = \sum_{n=b+1}^{2b}A_n - \sum_{n=2}^{b}A_n$$ where $A$ is an infinite set of non-unique natural numbers
user1825464
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Can you give me a hint finding a sum of this arctangent?

Well, I am trying to solve find a sum of arctan, but can't find a way. Can somebody give me a hint? $\Sigma^\infty_{n=1}tan^{-1}(n+1)-tan^{-1}(n)$ I have tried to integrate it but it seemed way too complicated. I hope there is another way. The…
강승태
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Find the sum of $2^{-x}/x$

Task is the following: find the $\sum_{x=1}^{+∞} \frac{2^{-x}}x$ I don't even know how to proceed. I know that $\sum_{x=1}^{+∞} 2^{-x} = 1$. However, is it useful here?
user13
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General formula for $\sum_{i=1}^{n} i^pA^i$

I was trying to sum series for fun and then I thought that since I can find the sum of polynomials and that of a geometric series then maybe I can find the sum of their product..I could find $$ \sum_{i=1}^n iA^i $$ Where $A$ is a constant easily…
Km356
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Interchanging the order of summation

I want to compute this sum: $$\sum_{S\,\subseteq\,Q} f\left(S\right)$$ where $Q$ is some finite set with $n$ elements. I think the first step should be: $$\sum_{i=0}^{n}\left(\sum_{S\,\subseteq\,Q\,;\,\vert S\vert\,…
augustoperez
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Is the following reasoning correct?

Working on a proof right now and would like to use the following reasoning: Given a set $R \subset \Bbb N \times\Bbb N$ and a number $b \in \Bbb N$. If we know that there is a subset $R_1 \subset R$ such that $$\sum_{(a_j, a_k) \in R_1} a_j \le b \…
Julian
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summation involving floor of terms of a GP

This is an interesting question i came across but am unable to find a solution to it. Given $N$, $k$, we need to find $M$ such that $M+\left\lfloor \dfrac{M}{k} \right\rfloor+ \left\lfloor \dfrac{M}{k^2} \right\rfloor + \ldots + \left\lfloor…
Aman Agarwal
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Evaluation of a series by hand

I want to evaluate the series $$\sum_{n=0}^{\infty}\frac{nx^n}{(1+x)^{n+1}}$$ I don't know how to do this by hand, but Mathematica tells me that the answer is: $$\sum_{n=0}^{\infty}\frac{nx^n}{(1+x)^{n+1}}=x$$ and this is indeed the answer I…
Quantumania
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Summation of product of integers with consecutive integers as first $m$ factors

It is well known that $$\sum_{r=1}^n r^\overline{m}=\frac {n^{\overline{m+1}}}{m+1}\\ \text{i.e.} \sum_{r=1}^n \scriptsize(r+1)(r+2)\cdots (r+m-1)=\frac{n(n+1)(n+2)\cdots (n+m)}{m+1}\\ \text{which can also be written as }\\ \qquad \scriptsize…
Hypergeometricx
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