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 compute this finite sum $\sum_{k=1}^n \frac{k}{2^k} + \frac{n}{2^n}$?

I do not know how to find the value of this sum: $$\sum_{k=1}^n \frac{k}{2^k} + \frac{n}{2^n}$$ (Yes, the last term is added twice). Of course I've already plugged it to wolfram online, and the answer is $$2-\frac{1}{2^{n-1}}$$ But I do not know how…
Don
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Infinite series of alternating reciprocals $\frac 1{1\cdot3}-\frac 1{3\cdot 5}+\frac 1{5\cdot 7}-\cdots $

Having misread the recent question here as $$\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)(2n+3)}$$ and having solved it, I thought that I would post it as a question instead. It has a rather interesting answer! Edit Now that we have nice solutions from…
Hypergeometricx
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Can $s_n=1+x+x^2+x^4+x^8+\ldots+x^{2^n}$ be simplified?

Is there a closed form for a finite sum $s_n=1+x+x^2+x^4+x^8+\ldots+x^{2^n}$? I apologize if this has been already discussed. Is there some literature on this I can take a look at?
Poppy
  • 866
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Summation of simple sequence

This is a fairly simple question, I'm sure, but I appear to be having trouble. What is the result of the following sequence: $$\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+ .... + \frac{n}{2^{n}}.$$ ? Thanks
Jojo
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How to compute the sum $\displaystyle\sum_{n=0}^{\infty}nP_{n}$

Following Problem is from probability theory:Define $G(n),P(n)\ge 0,n\in\mathbb{N}$,and such $$\begin{cases}G(n)=e^{-\lambda}\cdot\dfrac{\lambda^n}{n!},\lambda>0\\ \displaystyle\sum_{j=0}^{n}G(j)P(n-j)+(n+1)[P(n+1)-P(n)]=0,\forall n\in \mathbb{N}…
user237685
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Summation stuck under radical sign

I am trying to evaluate the following sum, but I'm unable to solve it in any general way. $$S=\sum_{k=1}^n\sqrt{1+\frac{1}{(k)^2}+\frac{1}{(k+1)^2} }$$ How can I do it?
TIWARI
  • 757
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Other variation of Nicomachus's Theorem?

We all know that $ 1^3+2^3+3^3 + \ldots + n^3 = (1+2+3+\ldots + n)^2 $. Denote $\displaystyle S_m = \sum_{j=1}^n j^m $, then we can set $ S_3 = S_1 ^2 $ for all positive integers $ n $. Question: Is there any other solution for the equation $ S_a=…
GohP.iHan
  • 1,376
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Simplification of a double sum involving partial sums of harmonic series

Could somebody explain the jump in the following equation? $$\frac{1}{n}\sum\limits_{i=1}^{n}\left[1 + \sum\limits_{j=i+1}^{n}\frac{1}{m}\right] = 1 + \frac{1}{nm}\sum\limits_{i=1}^{n}(n-i) $$
Joel
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Summation Notation: $i

Is this true? $$ \sum\limits_{i
Benck
  • 143
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How do I evaluate $\sum_{r=1}^{n} [r(r+1)(r+2)(r+3)] $?

I need to evaluate: \begin{equation} \sum_{r=1}^{n} [r(r+1)(r+2)(r+3)] \end{equation} But I don't know where to start. Can someone possibly give a hint? I need to solve this using telescopic series.
user63762453
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Summation of trigonometric functions such as $\sin x$

I am currently studying Integration (a very basic introduction) and I have a question regarding the summation of trigonometric functions. Given $f(x) = \sin x$, determine the area under the curve between a and b. By definition of a definite integral…
Shane M
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Calculating sums

My maths teacher showed me something on how to calculate sums. Let's take an example: $$\sum_{k=1}^n k(k+1) = \sum_{k=1}^n k^2 + \sum_{k=1}^n k = \frac{n(n+1)(2n+1)}{6} + \frac{n(n+1)}{2} = \frac{n(n+1)(n+2)}{3} $$ This was an easy one, but I just…
Victor
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Converting Summation to Expression

How does the summation break down from $$\displaystyle\sum_{n \geq 0} (x + x^2) ^ n$$ to $$\frac1{1 - x - x^2} $$ per this answer?
Kevin Meredith
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When does sum to infinity starts getting negative?

There is a popular claim going around the internet that the sum of the positive integers is $-1/12$. There are proofs to this statement and I am not going to try and refute them. But I could not understand at what point the sum starts becoming more…
Standstill
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Calculate summation of square roots

Calculate summation of square roots i.e $$\sum_{i=1}^N\sqrt{i}$$ I tried to search for its formula on the net but I couldn't find any of its sources.
typedef
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