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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Summation distributive definition

Suppose I have the following definition: $$\sum_{k=0}^N m_k\zeta^{1-k},$$ where $m_k$ is an integer and $\zeta$ is a complex number $\zeta=e^{i\theta}=\cos \theta + i\sin\theta$ with $\theta = 0\cdots2\pi$. This can also be expanded…
BeeTiau
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Double Summation $\sum_{i=1}^{n+1}\sum_{j=0}^{n-1}(i+j)=(2n+1)\sum_{r=1}^n r=3\sum_{r=1}^n r^2$

It can be easily shown by step-by-step and rather messy summation over $j$ and then over $i$ that $$\sum_{i=1}^{n+1}\sum_{j=0}^{n-1}(i+j)=\tfrac12 (2n+1)n(n+1)$$ Note that RHS is equivalent to $$\displaystyle (2n+1)\sum_{r=1}^n r$$ (1) Is there a…
Hypergeometricx
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Solving for a term in a summation

How can you solve for the summation term, $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n\pi)^2}$, in the Fourier series below? $$ x^2 \thicksim \frac{1}{3} + 4\sum_{n=1}^{\infty} \frac{(-1)^n}{(n\pi)^2}\cos{n\pi x} $$ After rearranging the terms, I'm stuck…
EarthIsHome
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What is the difference between these two sum notations?

I often see sum-notation wirtten in one of the following ways: AND: Is there any difference between the two, or is it just a matter of style? thanks.
Daniel Valland
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How can I calculate this summation? $\sum_{x=60}^{100} {100\choose x} $?

How can I calculate this summation? $$\sum_{x=60}^{100} {100\choose x} $$ ? I don't have idea how to calculate it, I tried to arrive at a probability expression of a random variable that is binomial ($Bin(n,p)$) but But I did not succeed.
AskMath
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How can I calculate this sum $\sum_{x=1}^{\infty} x^2\cdot q^{x-1} \cdot p$?

How can I calculate this sum (while $0 < q,\ p < 1$)? $$\sum_{x=1}^{\infty} x^2\cdot q^{x-1} \cdot p$$ I thought to calculate it with the derivative of something that gives $x^2\cdot q^{x-1} \cdot p$, but I don't know how to do it.
AskMath
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Summation notation with no starting \ stopping point

I'm work through some questions relating to connection coefficents. My question is more about the summation notation being used. Why is there no starting point (or end point) for the summation here? For the i, j, k, I take these to range from 1 to 2…
discojoe
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simplifying a summation related to an epidemic process

Here is a question that I came across while trying to understand some fundamental mathematical concepts related to the spread of epidemics. The problem turns out to be a mathematical puzzle related to the binomial coefficient. Consider an epidemic…
Daniel S.
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Show that the sum is less then $O(2^{n^{\epsilon}})$ for some $0<\epsilon <1$

Here is a problem. Say $n= 2^k$ for some $k\in \mathbb{N}$. Find the smallest $\epsilon>0$ such that $$2\cdot 2^{n\over 2}+ 2^2\cdot 2^{n\over 4}+...+2^k\cdot 2^{n\over 2^k}<2^{n^{\epsilon}}$$ for sufficiently large $n$. My work: I managed to…
nonuser
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How to prove the general formula for the Basel problem?

So I understand Euler's proof of the Basel Problem: $$\sum_{r=1}^{\infty}\frac{1}{r^2}= \frac{\pi^2}{6}$$ But how would I prove the general formula? Is it possible to prove without going into complex calculus and Fourier…
mathkid225
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find the sum of the $\sum_{k=1}^{2n}\frac{(-1)^{k-1}}{a_{k}}$

Let $a_{k}=\binom{2n}{k}$,find the sum $$\dfrac{1}{a_{1}}-\dfrac{1}{a_{2}}+\dfrac{1}{a_{3}}-\dfrac{1}{a_{4}}+\cdots+\dfrac{1}{a_{2n-1}}-\dfrac{1}{a_{2n}}$$ $A:\dfrac{1}{n+1}$ $~~$…
math110
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Properties of an interesting summation $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} }{2k+1}$

Here I found a summation from somewhere : $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1} }{2k+1}$$ Is it a famous summation? I've heard that it's a "special summation and irrational". Can anyone provide me a proof?
Mathejunior
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Confusion in proof regarding the summation operator

Why does $\sum_{n=0}^\infty\left[\sum_{m=0}^n\left[\frac{1}{m!(n-m)!} \cdot A^{n-m} \cdot B^m\right]\right]=\sum_{l=0}^\infty\left[\frac{1}{l!} \cdot A^l\right] \cdot \sum_{m=0}^\infty\left[\frac{1}{m!} \cdot B^m\right]$ hold? What properties are…
Anna D.
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Sum over Subadditive functions

Suppose $f(x)$ is a non-decreasing sub-additive convex function In order words, $f(x+y)\leq f(x)+f(y)$ for all $x,y$ and $f(x)\leq f(y)$ if $x\leq y$ Let $x_1,x_2\ldots x_i$ are $i$ positive integers such that their sum is $n$. What will be the…
Vk1
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Show that $\sum\limits_{r=1}^n\sum\limits_{k=1}^{n-r+1}(2r-1)k=\frac 13\sum\limits_{r=1}^nr\sum\limits_{s=1}^{n+1}s$

Question: Show that $$\sum_{r=1}^n\sum_{k=1}^{n-r+1}(2r-1)k=\frac 13 \sum_{r=1}^nr\sum_{s=1}^{n+1}s$$ purely by manipulating summation limits and summands, i.e. convert the original summation (on the left) to the summation on the right,…
Hypergeometricx
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