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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Summation problem (not sure of my solution)

If $f(x)=3-x$, evaluate $\sum_{k=1}^3f(2k)$. So this is my solution: $f[2(3-1)]=4$ $f[2(3-2)]=2$ $f[2(3-3)]=0$ So the required sum is $4+2+0=\boxed{6}$ Is my solution correct?
banana
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Is there a difference between these two sums?

Let $i,j \in N$ with $i \neq j$, and let $t \in T$. The sum is written as : $$\sum_{j \in N \setminus\{i\} }(x_{i,j,t} + x_{j,i,t}) = 2 \qquad \forall i \in N, t \in T$$ Would this be the same as writing: $$\sum_{t \in T}\sum_{i \in N} \sum_{j \in N…
mathplzfun
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Compute $\sum_{n=1}^{\infty}\left(\frac{\sin(n)}{n}\right)^2$

How would you evaluate $$\sum_{n=1}^{\infty}\left(\frac{\sin(n)}{n}\right)^2$$ Wolfram|Alpha says it equals $\;\pi/2-1/2.\;$ If the solution is complicated, ... I can handle complicated.
Royce Martin
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A little confusion in this step with two finite sums.

I was reading some examples and I got stuck understanding these steps: $$\sum_{m=1}^{100}\sum_{n=1, m\ne n}^{100}mn\frac{1}{9900}=\frac{1}{9900}\sum_{m=1}^{100}(m\sum_{n=1, m\ne n}^{100}n)=\frac{1}{9900}\sum_{m=1}^{100}(m(\sum_{n=1}^{100}n-m))$$ I…
Pwaol
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Expressing $\sum \sum_{i,j \in[-n,n], (i,j)\ne (0,0)} \frac{(-1)^{j+k}}{\sqrt{j^2+k^2}}$ in a different way

I try to separate the following double sum $$S=\sum_{i,j \in[-n,n], (i,j)\ne (0,0)} f_{ij}$$ where $$f_{ij}=\frac{(-1)^{j+k}}{\sqrt{j^2+k^2}}$$ $$S=\sum _{k=1}^n \sum _{j=1}^n (f(-k,-j)+f(-k,j)+f(k,-j)+f(k,j))$$ but I do not know why it does not…
user547221
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Kernel Inner Product

I am trying to follow some lecture notes about computing a kernel from the inner product of two features maps. I don't understand how this equation: $<\phi(x),\phi(z)>=1+\sum_{i=1}^{d}x_{i}z_{i} +\sum_{i,j\in\{1....d\}}x_{i}x_{j}z_{i}z_{j}…
486DX2-66
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Summation equation for products

So if you want to add up all of the numbers from 1 to 100, the equation is fairly simple: $ \frac{n(n+1)}{2} $ Does a similar equation exist for multiplying every number from 1 to 100 together?
Muffinlicious
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Calculate $\sum_{i=1}^{n-1} i\alpha^{2i}$

I try to calculate $\sum_{i=1}^{n-1} i\alpha^{2i}$ I think one week but still have no ideas to this sigma,thank everyone
Hakke
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Help with understanding sigma sum-related notation

My maths book uses this kind of notation for sums (circled in red) but doesn't explain what it means. I don't know what it's called or where to find resources explaining it. Could someone explain it to me or point out to somewhere where I can read…
user911422
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Double summation with dependent variables

Currently i have taken part in a coding contest where i was asked this question https://atcoder.jp/contests/abc186/tasks/abc186_d. In the editorial solution they have given something like this: Here especially in the second last step how the outer…
Harsh Srivastava
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Limit does not exist or limit does exist? What does the unevaluated output mean?

From this short Mathematica program while investigating the convergence of the Dirichlet series for the Möbius function: Clear[a, b, s, x]; s = 1/2 + 100*I; Limit[1 - Sum[1/a^s, {a, 2, x}] + Sum[Sum[1/(a*b)^s, {a, 2, x}], {b, 2, x}], x ->…
Mats Granvik
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Boyd & Vandenberghe, exercise 3.19(a) — I'm a bit confused

I was working through the exercises of chapter 3 from Boyd & Vandenberghe's Convex Optimization but I'm a bit confused by the solution for Ex 3.19(a). It says that:- for $\alpha_1 \geq \alpha_2 \geq \alpha_3 \geq ... \geq \alpha_r \geq 0$ and…
Hazard
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Evaluating $\sum_{k>\frac{N}{2}}\frac {1}{N}\cdot \frac{N-k}{k}$

Assuming $N$ is even, how can I evaluate the following sum: $$\sum_{k>\frac{N}{2}}\frac{\binom{N}{k}(k-1)!(N-k)!}{N!}\cdot\frac{N-k}{N}= \sum_{k>\frac{N}{2}}\frac {1}{N}\cdot \frac{N-k}{k}$$ I really don't know how to do it... Thanks! (Not HW BTW)
Amihai Zivan
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How to prove that 1/1!+1/2!+1/3!+....<3?

How can I prove that 1/1!+1/2!+1/3!...<3? Should I use mathematical induction? I can see that the sum is the same as the power series of e^1 and that is also less than 3. But where is the proof for that?
sakib hasan
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Double sigma formula with nested index

I am trying to obtain a formula for: $G = m_1m_2 + m_1m_3 + m_2m_3$ using sigma notation. I've come up with: $\sum_{i=1}^{3} \sum_{j=2}^{3} m_im_j = \sum_{i=1}^{3}m_im_2 + m_im_3 = m_1m_2 + m_1m_3 + m_2m_2 + m_2m_3 +m_3m_2 + m_3m_3$. My question is,…
user856485
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