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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What did this sum become: $\left(\sum_{n=a}^{b} z_n\right)^2 $?

Can I see something meaningfull about this sum? Where is it equal to? $$\left(\sum_{n=a}^{b} z_n\right)^2$$ Is it equal to: $\sum_{n=a^2}^{b^2} z_n^2$ or something else, I've no idea how to deal with it. Thanks for the help.
Jan Eerland
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Checking presentation of work

I am evaluating the following summation, $$\sum_{r=0}^9 (r^3 - 1)$$ and I have gotten $$\sum_{r=1}^9 r^3 - \sum_{r=1}^91 = \frac{1}{4} \cdot 9^2 \cdot (9+1)^2 - 9 = 2016$$ Is this the correct way to express my solution? Please correct me if there…
Lily L
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How to find the sum of a sequence, e.g. $(2n - 1)^3$

I need to find the sum of this series: $1^3$, $-2^3$, $3^3$, $-4^3$, ... ,$2n^3$ so I've split that up into: $\sum_{r = 1}^n (2r-1)^3 - \sum_{r = 1}^n (2r)^3$ So all I need is how to find: $\sum_{r = 1}^n (2r-1)^3$
Chris Fullerton
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Find the value in terms of $n$

$$\sum_{k=0}^{n} 3^k \binom{n}{k}$$ I have tried filling in and simplifying but the following was not correct: $$1 + 3n + 3^2 \dbinom{n}{2} + \cdots + 3^n$$
Anonymous
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Evaluate $\sum_{m=1}^{\infty}p^{m-1} e^{\lambda z^m}$

I've spent hours on a problem, but it boils down to evaluating this sum: $\sum_{m=1}^{\infty}p^{m-1} e^{\lambda z^m}$ where, $z, \lambda\in \mathbb{R}, p\in [0,1]$ I don't know how to evaluate it. Is this trivial?
Kerry
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Can somone help me do this double sum problem. I know how to do it manually, but I would like to know how to do it using summation formulas.

Calculating the double sum: $$\sum\limits_{i=1}^{10}\sum\limits_{j=0}^{15}(3i+2j)$$ I know how to do this manually, but I would like to know how to do it using a summation formula. Could somone please show me the steps as to how I would be able…
Nwqp
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Why is $(1+2+3+4+...+n)^2$ equal to $1^3+2^3+3^3...+n^3$?

I noticed that the sum of the first $n$ cubes is equal to the square of sum of the first $n$ natural numbers: $$ \sum\limits_{i=1}^n i^3=\frac{n^2(n+1)^2}{4}=\left(\frac{n(n+1)}{2}\right)^2=\left(\sum\limits_{i=1}^n i\right)^2 $$ Is there a clever…
gonthalo
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How to find the sum $\frac{1}{1^4+1^2+1}+\frac{2}{2^4+2^2+1}+..+\frac{2015}{2015^4+2015^2+1}$?

How to find the sum $\frac{1}{1^4+1^2+1}+\frac{2}{2^4+2^2+1}+..+\frac{2015}{2015^4+2015^2+1}$ ? I'm not being able to approach the problem.Hints please!
user220382
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Find the sum of the n terms of the series $2\cdot2^0+3\cdot2^1+4\cdot2^2+\dots$

Find the sum of the n terms of the series: $2\cdot2^0+3\cdot2^1+4\cdot2^2+\dots$ I don't know how to proceed. Please explain the process and comment on technique to solve questions of similar type. Source: Barnard and Child Higher Algebra.…
gaufler
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Surprising Summation (3): $\sum_{i=1}^n\sum_{j=1}^i 2(n-i)+1=\sum_{i=1}^n i^2$

Show that $$\sum_{i=1}^n\sum_{j=1}^i 2(n-i)+1=\sum_{i=1}^ni^2$$ without expanding the summation to its closed-form solution, i.e. $\dfrac 16n(n+1)(2n+1)$ or equivalent. Background as requested: The summands for both equations are not the same but…
Hypergeometricx
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Sigma Notations

I have troubles understanding the sigma notation. If for example we have $c_i$ as $$c_i=\frac {x_i-x}{\sum(x_i-x)^2}$$ $$\sum c_i=\sum\frac{x_i-x}{\sum(x_i-x)^2}$$ Do we distribute the sigma to both top and bottom? But then the bottom would have…
Skipe
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Changing indices of a sum (Basic question)

I was working through some proofs and it came to a point where I had to change the index of a sum which started as sigma (from k=0 to n-1) Substituting in i=k+1 So that means, my k=0 would become i=1 But does that mean I also have to increase the…
sahimat
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How to interpret double summation with same subscripts

How would I interpret the following summation (where $r_{k}$ is a function that produces a scalar): $f^{int}_i = \frac{(\kappa \mathbf{b})_{i} + (\kappa \mathbf{b})_{i + 1}}{2} \displaystyle\sum_{k=i+1}^{N-2} [r_{k} - \frac{(\kappa \mathbf{b})_{i -…
stagermane
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Adding matrix elements using single summation

Is it possible to add all elements of square matrix using single summation notation, instead of using double summation?
Dilya
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What is the sum of integers between 1 and 200 inclusive that are divisible by both 4 and 5?

What is the sum of integers between 1 and 200 inclusive that are divisible by both 4 and 5? Now I attempt the question of taking the integers to be divisible by 20(LCM of 4 and 5) and get the number of terms as 10 and the sum as 1100. Is it the…
Jason
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