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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Series with denominator that is square of product of consecutive numbers

Evaluate $\sum_{k=1}^{\infty} \frac{1}{k^2(k+1)^2}$. I've tried to use a telescoping approach, but it doesn't seem to work. It doesn't also seem to converge to a "pretty" value as well.
The Math Troll
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Evaluating $\sum_{i=2}^7\,2^{i+2}$

I am going through my textbook and have come across this question that asks to work out the sigma, this sigma is troubling me, I am not understanding how to do the "to the power of I + 2" part. Any help would be great,…
Richard Motion
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How to get an approximative behaviour of an alternating sum

I have an alternating sum and am looking for its asymptotic behaviour. Due to the change of signs there occure a lot of cancellations between the different terms. However the terms are all different, so no terms cancel exactly. Is there still a…
user404302
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Show that $s_z^2 = s_x^2.$

Let $x_1, \ ... \, x_n \in \Bbb R$ and $b \in \Bbb R$. Assume that $z_k = x_k + b, \ k = 1, \ ... \, n$ and $\bar z = \bar x + b$. Show that $s_z^2 = s_x^2.$ Here, $$\bar x = {1 \over n} \sum_{k=1}^n x_k$$ and $$s_x^2 = {1 \over n - 1}…
Borol
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Simplification of a sum with double index

Im trying to simplify the sum $$ S_n = \sum_{1 \leq j < k + j \leq n} \frac{1}{k} $$ Attempt: I can write $$ (1 \leq j < k + j \leq n ) = (1 \leq j \leq n) \cap(j < k + j \leq n)= (1 \leq j\leq n) \cap (0 < k < n-j)$$ Thus, $$ S_n = \sum_{j=1}^n…
ILoveMath
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What is the summation in this equation?

$$\frac {\sum_{i=1}^N (x_i-u)/N}{p^4}$$ Is it only top part of the fraction including the N or just whatever is in the parenthesis. I'm making a code that includes this equation and I need to put it in a loop so that is why I am asking. Also…
Rafael
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Plot color space

I am trying to plot the following shape: I am starting with the following equations, then converting from XYZ to xyY: http://www.brucelindbloom.com/index.html?Eqn_Spect_to_XYZ.html Here is my POV-Ray code, which is producing wrong results: #local…
posfan12
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Why is $\sum_{j = 1}^{n} \dfrac{1}{n} = 1$

I encountered $\sum_{j = 1}^{n} \dfrac{1}{n} = 1$ in my textbook, but I do not understand why this summation $= 1$. My textbook provides no reasoning as to why this is the case. My understanding is that, since there is nothing in $\dfrac{1}{n}$…
The Pointer
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$\sum_{\mathbf S\in\{-1,1\}^n}\sum_{i=1}^{n-1} S_iS_{i+1}$ where $\mathbf{S}=\langle S_1, S_2, \ldots, S_n\rangle$

I'm trying to figure out the value of this sum, and if I'm correct it appears to be 0. Check please this solution. So we have $n-1$ elements ($S_iS_{i+1}$), and every element can be $1$ or $-1$. Because if we put in first element 1 (for example $S_1…
Eugene Korotkov
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How to solve a summation equation for the number of times added

So I have the equation https://puu.sh/uosqs/d90b94c2f5.png $\sum_{i=0}^{h}m^i=v$ and I need to solve for $h$. How do I rearrange this equation so that it is in the form $h$ = something?
Malingos
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Better way to show that $a^{n-1}+a^{n-2}+\dots+a+1 \neq 1$

Better way to show that $a^{n-1}+a^{n-2}+\dots+a+1 \neq 1$ The way that I am currently trying is to show that $$a^{n-1}+a^{n-2}+\dots+a+1 \neq 1$$ $$a^{n-1}+a^{n-2}+\dots+a \neq 0$$ I then follow with a wishy washy argument about how if a is…
MathIsHard
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Simplification of a special iterated sum

Is it possible to simplify this iterated sum $$f(x)=\sum_{i_{u-1}=u}^{x}\sum_{i_{u-2}=u-1}^{i_{u-1}} \cdots \sum_{i_2=3}^{i_3} \sum_{i_1=2}^{i_2} \sum_{k=1}^{i_1} 1$$ by using a binomial coefficient? Is this iterated sum still equivalent of…
Dingo13
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Help expanding this summation for VDW mixing rules

I've hit a bit of a wall when trying to expand this summation but for the life of me can't get it right. $\sum_{i} \sum_{j} y_i y_j {a_i}_j $ for a 2 component system.
SimpleJack
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How do I solve this? Sum notation, sums

When $\sum^{60}_{r=0} 2^r = 2^n-1$ what is n? I forgot to mention that I can't use a calculator on this
Elias
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How would you write this series into a for loop?

$$\sum_{i=1}^5 (-1)^{i+1}*i^2 $$ I thought it would be something like this but i always get 36 or -36. This is c also. for(i = 1; i <= n; i++) if(i++ % 2 == 0) { count = pow(-1,n+1)*pow(n,2); } else{ count =…
billnye852
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