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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Trouble summing $\frac{1}{3^i}$ from 1 to n

$\sum_{i=1}^n \frac{1}{3^i}\tag{displayed}$ I can't figure this out. I expanded it: $(\frac {1}{3^1}+\frac{1}{3^2}+...+\frac{1}{3^n})=S$, and I think the technique is to multiply both sides by something and then subtract, but I'm not quite sure how…
Rokko
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Are these two sums equal?$\sum\limits_{i=1}^{n^2}\sum\limits_{j=i}^{i+n}\sum\limits_{k=1}^j = \sum\limits_{i=1}^{n^2}i+(i+1)+(i+1)+...+(i+n)$

$\sum\limits_{i=1}^{n^2}\sum\limits_{j=i}^{i+n}\sum\limits_{k=1}^j = \sum\limits_{i=1}^{n^2}i+(i+1)+(i+1)+...+(i+n)$ Are these two sums equal? If n = 2 so the result of $\sum\limits_{i=1}^{n^2}\sum\limits_{j=i}^{i+n}\sum\limits_{k=1}^j$ is equal 30…
Michael
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How to simplify this power equation?

I need a function such that if input is $n$ it outputs an $n$-digit number. I need a mathematical function. For example: $$\begin{align}\mathrm{ANY}(2) &= 22 \\ \mathrm{ANY}(3) &= 333 \\ \mathrm{ANY}(n) &= nnn\ldots n \quad (n \text{…
Grijesh Chauhan
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Changing limits of summation

I have $\sum_{n=0}^\infty \frac{{(-1)}^n}{z^{n+1}}$ now I want to change the lower limit to begin at $1$ instead of $0$. So I let $k=n+1$, and I'm supposed to get $\sum_{k=1}^\infty \frac{(-1)^{k+1}}{z^k}$. I know how to get it for $z$ because I…
A A
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Is there any. elementary formula for the sequence$\sum_{k=1}^{n}\left(2k-1\right)\left(\frac{1}{2}\right)^{k}$

Is there any formula for the following sequence which does not use any derivative and also is less advanced: $$\sum_{k=1}^{n}\left(2k-1\right)\left(\frac{1}{2}\right)^{k}$$ I've calculated the general formula and here is a photo of the solution …
Absurd
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Split up sum of products $\sum{a_i b_i}\approx(1/N)\sum{a_i}\sum{b_i}$ for uncorrelated summands?

As the topic says, is $\sum{a_i b_i}\approx(1/N)\sum{a_i}\sum{b_i}$ possible when $a_i$ and $b_i$ uncorrelated? I have come across something like that very recently where this has been magically done without additional explanation and it makes…
Majte
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Property of a square made of XOR of the values of its zero indexed columns and rows

Hi please I need some help with this : Given this square generated by XORing the values of indices of rows and columns (I believe it's called munching squares (check picture) let's take for example the black rectangle: I noticed that the sum of…
KamuiYato
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Rewriting sum using summation notation, then finding the sum

I'm supposed to rewrite the sum 1,3,7,15,31...1023 using summation notation, then find the sum. I can tell the value being added begins with 2 then doubles to 4,8,etc. But I'm not sure how to write it in summation notation, I figure it's some sort…
JCHLW
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Computing the sum $\sum_{n=1}^\infty\frac{1}{\left\lVert\vec{r}_n\right\rVert^4}$

I try to compute the sum of the inverse distances to the fourth power of all unit cubes inside $\mathbb R^3$: $$\sum_{n=1}^\infty\frac{1}{\left\lVert\vec{r}_n\right\rVert^4}$$ with cartesian coordinates this sum…
Caesar.tcl
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what is the value of $\sum_{ijk}\omega u_iu_j \delta_{ik}\delta_{kj}$?

I have to work with the following sum: $\sum_{ijk}\omega u_iu_j \delta_{ik}\delta_{kj}$ where $\omega$ is a constant in $\mathbb{C}$. Is the answer: $$\omega \sum_k \sum_i u_i \delta_{ik}\sum_j u_j\delta_{kj}=\omega\sum_k…
Matt
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Programming code with BASIC

I have to program in BASIC a code to do this: From a collection of 10,000 cannonballs, a square based pyramid is built with a single cannonball on top and a square number on each layer. How many layers can be made? How many cannonballs are left…
Santiago Baigorria
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Nested-Recursive Summation

How could I simplify the following ? $$ {\large\sum_{k_{1} = z}^{n}\sum_{k_{2} = k_{1}}^{n}\sum_{k_{3} = k_{2}}^{n} 1}\qquad \mbox{for natural numbers}\ z, n $$
user_hello1
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How to express a conditional in a summation

I have a question about notation: What's the proper mathematical notation to write (sum from i=1 to X)(sum from j=1 to X) 1 if gcd(i,j)==2, 0 otherwise?
Sean Hill
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Question About Summation of Summations

I am trying to figure out the value of $$\sum_{i=2}^4\sum_{j=2}^i j$$ I thought of having a two for loops, one nested inside of the other, so: for(int i = 2; i <= 4; ++i) { for(int j = 2; j <= i; ++j) { sum += j; } } So I got: i =…
Tim Weah
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Sum of the sum of the sum of $4i(-1)^i$

I am attempting to evaluate the sum $$ \sum_{i=0}^{x-1} \Biggl(4+4\sum_{j=3}^{i+2} j(-1)^j\Biggr) $$ I'm not that experienced at summations, so I can go as far as $$ 4x+4\sum_{i=0}^{x-1} \sum_{j=3}^{i+2} j(-1)^j $$ Before I get stuck. Using a…
Jonah
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