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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Multi-dimensional summation operator

For example we could simplify $$\int_{x\in X}\int_{y\in Y}fdxdy$$ To $$\int_{v\in A}fdv$$ Can we simplify the summation operation $\sum_{i\in \mathbb N}\sum_{n\in\mathbb N}a_{in}$ in a similar way? I heard about symbols such as $\sum\oplus$ but I am…
High GPA
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Hints on solving $\sum_{i=1}^n i \cdot 2^{i-1}$

I need hints on solving this summation. Problem $\sum_{i=1}^n i \cdot 2^{i-1} = \ldots$ Thanks in advance.
Muhamad Abdul Rosid
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How to solve $\sum_{i=a}^b\sum_{j=i}^c(j-1)$

I really have looked for about 2 hours on the Internet, but I cannot find how to solve $\sum_{i=a}^b\sum_{j=i}^c(j-1)$. Can somebody please help?
Faceb Faceb
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Simplify summation into closed form

Have no idea where to go. A friend suggested taking the difference of Gauss sums, not sure what that means: $$\sum_{i = 1}^{n}\sum_{j = i}^{2i}j$$
user7252850
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Bordered Hessian. I have no idea.

On plane $OXY$ find point $P(x,y)$, for which sum of squared distances to given points $P_{i}(x_i,y_i)$, $i=1,2,...,n$ is the smallest possible.
Nowak Grzegorz
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Please help me calculate this sum

\begin{equation} \begin{split} \sum_{j=1}^{p}\left[\frac{1}{\frac{1}{p-\alpha(j-1)}(1-\frac{1}{p-\alpha(j-1)})^{p-j}} \cdot\frac{1}{(\theta-\frac{1}{p-\alpha(j-1)})(p-j+1)}\right] \\ \end{split} \end{equation} $p$ is a finite constant, $\theta$ is a…
Zhen Cao
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Prove Finite Sum

To prove (1) $$\sum_{i=1}^n \dfrac {i}{2^i} <2$$ Someone shows that (2) $$\sum_{i=1}^n \dfrac {i}{2^i} = 2-\dfrac{n+2}{2^n}$$ so we can prove (1). Can anyone guide me how from (1) they come up with (2) please ? On what base they find (2) ?
xuoimai
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simplify the summation functions

I have the following equation and I am trying to simplify it: $$\frac{\sum_{i=1}^M \sum_{j=1}^N [f(i,j) - h(i,j)]^2}{\sum_{i=1}^M \sum_{j=1}^N [f(i,j)]^2}=1-\frac{2\sum_{i=1}^M \sum_{j=1}^N f(i,j)h(j,j)+\sum_{i=1}^M \sum_{j=1}^N…
Sarmad
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how to calculate multiple summations within each other?!!

I'm not sure how to calculate the attached equation with these multiple summations implemented into each other...... the variables a, b1, b2 in the last term include all the counters i1, i2 j1, j2: my idea is as follows in the attached img
F.Abdullah
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how to calculate summation with zero upper and lower boundaries?

Can i evaluate this if both upper and lower boundaries turn out to be zero?!! does it have a value or should i just disard it during calculation
F.Abdullah
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Summation $k=n-4$ to $k=4$ of $1$

This is perhaps very basic but I am currently very lost on how to think in order to end up with the answer: $9-n$ in the following summation: $$\sum_{i=n-4}^4 1 = 9-n $$ My first idea was to rewrite the summation with something like: $5+…
user469352
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Solving an equation for a sum index

I have the following equation: $$T=P\left [1+\sum_{i=1}^{n}(1.02)^i \right ]$$ How can I solve the equation for the $n$ value? I don't know how to expand the sum so I can handle algebraically $n$
Joshua Salazar
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Can we simplify this summation?

Is it possible to simplify the following expression in a form without the sum? $$\sum_{i=0}^n(e^x)^i(e^y)^{i^2}$$ equivalent to $$\sum_{i=0}^na^ib^{i^2}$$ where $a=e^x, b=e^y$ If it's not possible, is it nevertheless possible to do so if we take…
user56834
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How would I express a nested summation of n layers without expressing each individual layer?

If there is a nested summation with n layers, each layer having the same starting and ending value as every other layer, and the innermost layer containing some expression, how would I express this nested summation without expressing every…
Anders Gustafson
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Calculating summation on integer numbers from $-\infty$ to $\infty$

Does anyone know how to calculate the following summation, in which $a$, $b$ and $k$ are constant real numbers: $$\sum_{n=-\infty}^{\infty} \frac{k n +a}{\big((k n +a)^2 + b^2\big)^{\frac{3}{2}}}$$ In the above relation, $n$ is an integer number…
sara nj
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