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.

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changing summation limits

Let us say we have the following summation $$\tag 0 \sum_{k=0}^{\infty }\sum_{l=0}^{\infty }g_{k}h_{l}\delta (t-(l+k)T)$$ Now, we let $n = l + k$. Then $l = n - k$ which becomes $$\tag 1 \sum_{n=0}^{\infty }(\sum_{k=0}^{n }g_{k}h_{n-k})\delta…
vvavepacket
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Finding the right expression of a SUM

I would like to find an expression where one adds to $\gamma$ smaller and smaller fractions of $1/\gamma$, like: $p=\gamma+1/\gamma+1/\gamma^3+1/\gamma^5+...+/\gamma^n$ where n is always odd. I tried $\gamma+\sum_{i=1+2n}^n\gamma^{-n}$ But in…
Luthier415Hz
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Show $z\sum\limits_{n=0}^{\infty}(n+1) \left( \frac{1}{z^{2n}} \right)=\frac{1}{z}+\sum\limits_{n=1}^{\infty}(n+1)\left(\frac{1}{z^{2n+1}}\right)$

I am doing an exercise for which I have the sum below on the LHS and I am supposed to rearrange it to get the sum on the RHS. \begin{equation} z\sum\limits_{n=0}^{\infty}(n+1) \left( \frac{1}{z^{2n}}…
TK99
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How would I simplify a difference of a - b raised to the power 2m?

I want to simplify something of the form $$(a-b)^{2m}$$ in order to facilitate integration, so that it comes out of the form $$\sum(a^{2m-p}(-b)^p)$$ Please note that 2m is deliberately even by the result I got.
therickster
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Sum of order $\sum_{n=0}^{\infty}\frac{(-1)^n (n+1)}{(n+2)2^{n}}$

i need help solving this task, if anyone had a similar problem it would help me a lot. The task is : Find the sum of order $$\sum_{n=0}^{\infty}\frac{(-1)^n (n+1)}{(n+2)2^{n}}$$ I'm not asking anyone to do my task, I need a hint on how to start,…
LogicNotFound
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Explain to a 16 y.o. — Why $\prod\limits_{i\in I}(1+x_i)=\sum\limits_{J\subseteq I}\prod\limits_{j\in J}x_j$?

$\prod\limits_{i\in I}(1+x_i)=\sum\limits_{J\subseteq I}\prod\limits_{j\in J}x_j$– user940 Oct 1 '13 at 12:13 My child's 16. How can we intuit or prove this most easily? We prefer a Story Proof, rather than tedious algebraic manipulations or…
user851668
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Infinite sum over product of step functions

I am trying to evaluate the following sum $$S(m) = \sum_{n=0}^{-\infty}\theta(m+n)\theta(-n)$$ Where $m$ and $n$ are both integers and $\theta$ is a Heaviside step function. I am trying to understand whether $S$ has any finite value. Can anyone shed…
felix
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How do I read this sum? $\sum_{j \in N \setminus \{i\}} (x_{i,j,t} + x_{j,i,t}) = 1$

Let $N = \{1,2,3,4\}, T=\{1,2,3,4,5,6\}$ and let $i,j \in N$. I am trying to understand what the following sum means but am having a little difficulty in comprehending it: $$\sum_{j \in N \setminus \{i\}} (x_{i,j,t} + x_{j,i,t}) = 1 \qquad \forall…
mathplzfun
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How to simplify a summation using mathematica

I have the following summation I would like to simplify: $f(y)=\frac{1}{\left(2^{N}-1-N\right) \sqrt{2 \pi}} \sum_{k=2}^{N} \frac{\left(\begin{array}{l} N \\ k \end{array}\right)}{\sqrt{k \sigma^{2}+1}} e^{-0.5 y^{2} /\left(k…
wanderer
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How to properly solve this two summation sitting next with each other?

I just want to ask how do we solve this kind of summation? $$\sum_{i=1}^n x_i\sum_{i=1}^n y_i$$ I am confused how Do we solve for each summation first? then multiply the summation of x, to the summation of y? Or do we solve the summation of all…
himynameisjm
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How many ways can a number be summed to from known elements

In how many ways can I sum elements $a,b,c,d,...$ such that they add up to $n$? For example, $1,2,3$ can be summed to $4$ in $4$ ways because: $$4 = 1+1+1+1 = 2+2 = 1+3 = 2+1+1$$ If two ways use the same values but in other order (e.g. $1+2+1$) they…
pgp1
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Let $F$ be a finite subset of the natural numbers and consider the sum

$$\sum(-1)^l\tag{1}$$ Define $F_{\text{even}} = \{n \in F : n\text{ is even}\}$ and $F_{\text{odd}} = \{n \in F : n\text{ is odd}\}$ (a) Suppose that $\#F$ is odd. Show that $\#F_{\text{even}}\ne \#F_{\text{odd}}$ (b) Suppose that $\#F$ is odd.…
Felix Montoya
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Upper bound for $\sum_{n=1}^\infty \frac{2^n - n^3}{\sqrt{n!}}$

I am trying to find an upper bound to this expression $$ \sum_{n=1}^\infty \frac{2^n - n^3}{\sqrt{n!}} $$ It is sure that the limit of the inner expression as it goes to infinity is 0 (which is neccesary, not enough), but I can't find a way to show…
klaufir
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Relation between sum and cardinality of finite set

Let $A$ be a finite set with cardinality $\lvert A\rvert$. Then is it true that $\sum_{j\in A}1 =\lvert A\rvert$?
Miski123
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Could xᵢ = x̅ ???

I know it's a weird question. But this thing is confusing me. (x̅) : average μ      ① ∵ $\frac{1}{n}\sum\limits_{i=1}^n(x_i) = \bar{x}$ ∴ $\sum\limits_{i=1}^n(x_i) = {n}\bar{x}$      ② ∵ $\sum\limits_{i=1}^n(C) = {n}{C}$    | C = constant ∴…
Roo Tenshi
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