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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Decomposing a summation

How is it possible to go from 1 to 2? I can't include images in questions, so I've linked to them. $$R\cdot l\cdot\sum_{t=t_0}^{t_1}\frac{1}{L(t)}$$ $$R\cdot l\cdot\left(\sum_{t=0}^{t_1}\frac{1}{L(t)}-\sum_{t=0}^{t_0}\frac{1}{L(t)}\right)$$ From my…
curiousdweller
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Understanding summation decreasing index

I'me following some summation examples and I came to this situation $$|4-4| + \sum_{n=1}^{\infty} |4\cdot0.1^n| = -4+4\sum_{n=0}^{\infty} 0.1^n$$ How do they get to the last result? I thought that $|-4+4|=0$ and decreasing the index should become…
Favolas
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Find a closed form to $\sum_{k=0}^n 2^k3^{n-k}$

How would I find the closed form of $\sum_{k=0}^n 2^k3^{n-k}$ There's two properties of summations that I think apply here: $\sum_{k=1}^n a_kb^{i+k}=b^{i}\sum_{k=1}^n a_kb^{k}$ and $\sum_{k=0}^n a^k=\frac{a^{n+1}-1}{a-1}$ How do I apply them?
jem do
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How to sum this quasi-binomial summation without coefficients?

I have the summation $$\sum_{k =1}^{n + 2}x^{n + 2 -k}y^k$$ which expands out to (taking $n = 3$ as an example) $$x^4y^1 + x^3y^2 + x^2y^3 + x^1y^4.$$ How can I sum it? It looks similar to this question but the indices cannot be shifted to match…
user983799
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How to generalize a sum that relies on previously calculated terms

I have a sum that has the following form. $\underbrace{\underbrace{(7 + 2*1)}_\text{A} + {(7 + 2*A)}}_\text{B} + (7 + 2 * B) ....$ The first term is calculated for $n=1$, the second term for $n=2$ and so on. Each parantheses include the result of…
kklaw
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Summation with out limit

Can someone explain me how I read that sum notation? I have that task: $$n\in \mathbb{N}^* \quad and \quad A_1...A_n \quad are\ finite\ groups $$ $$ \#(\cup_{i=1} ^{n} A_i)=\sum_{I \subseteq[n], I\neq \emptyset}(-1)^{( \#I)}\#( \cap_{i\in…
MeepMeep
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Double Summation with double index

I have that question: $$ \sum_{i=1}^{n}\sum_{j=i+1}^{n} a_{i,j} $$ I try to understand how to sum this double summation works, so I try to sum it up to 3, but so far managed to find 3 different…
MeepMeep
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Find the sum of $\sum_{k=1}^{k=n} \frac{1}{k(k+2)}$.

Find the sum of $\sum_{k=1}^{k=n} \frac{1}{k(k+2)}$. Here is my solution. Since $\frac{1}{k(k+2)} = \frac{1}{2} \left(\frac{1}{k} - \frac{1}{k+2}\right)$, $\sum_{k=1}^{k=n} \frac{1}{k(k+2)} = \frac{1}{2}\sum_{k=1}^{k=n} \left(\frac{1}{k} -…
PRD
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Why is $2\sum_{k=0}^{n}4^k = \frac{2}{3}(2^{2k+2}-1)$?

I read it in a text saying they got there using standard techniques but I'm really rusty with this stuff because I've tried to prove it and don't seem to go anywhere.
kartzs96
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Find a simple Expression for a sum $t_n$

Context: Consider a sum; $$s_n= \sum_{m=1}^{n} m2^m$$ It has already been established previously that $s_n = f(n)$ where $f(n)=(2n-2)2^n+2$. Using this information find a simple expression for the sum; $$ t_n=n+2(n-1)+4(n-2)+8(n-3)+\ldots…
Swiss Gnome
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breaking a summation when the index is a factor

I am evaluating the sum $\sum\limits_{i=1}^{n}ip^i$. in this answer the following identity is used : $$\sum_{i=1}^n ip^i = \sum_{i=1}^n \sum_{j=i}^n p^j$$ I don't see where this comes from, maybe one could help me out ? As a computer science…
T.D
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Evaluating the sum $\sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (j - i)$

I've been working on trying to figure out the titular sum, $\sum_{i = 1}^{n} \sum_{j = i + 1}^{n} (j - i)$, which has an analytical solution by the Wolfram-Alpha output here. However, I have not been able to replicate this fully." line2 = "I began…
ndwadwads
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Strange behaviour of sums to infinity?

The following two sums I evaluated using mathematica $$ S_1 = \sum_{i = 0}^\infty \left(\frac{1}{4}\right)^i = \frac{4}{3} $$ and the sum $$ S_2 =\sum_{i = 0}^\infty \left(\frac{3}{4}\right)^i = 4 $$ This seems really strange to me as $S_2$ should…
Marten Folkertsma
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Why is $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{k}\geq \frac{1}{2}$?

I was playing around with adding fractions and noticed that when I start at any fraction of the form $1/n, n \in \mathbb{N}$, I get: $$\frac{1}{3}+\frac{1}{4}\geq \frac{1}{2},\; \frac{1}{4}+\frac{1}{5}+\frac{1}{6}\geq\frac{1}{2}$$ After trying out…
WaterDrop
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$\sum_{i=1}^n\sum_{j=1}^n{|i-j|}$

I was given this summation : $$\displaystyle\sum_{i=1}^n\sum_{j=1}^n{|i-j|}$$ And my idea was to use the fact that: $\displaystyle\sum_{j=1}^n{|i-j|}$=$\displaystyle\sum_{j=1}^i{(i-j)}$+$\displaystyle\sum_{j=i+1}^n{(j-i)}$ Any other solutions ?
Oussama Akar
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