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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Find the top $k$ smallest sums above $N$ from the collection $S$ - Duplicates allowed

Given a collection $S$ with up to $1,000,000$ values where $0 < S_i < 100$. I need to find the top $k$ groups whose sum is bigger than $N$ where $0 < k < 1,000$ and $N$ is a big positive number. The groups are sub-collections of $S$ where duplicates…
Ilya Gazman
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Problem in finding general term $(T_n)$ of given series

Question: What is the sum up to $n$ terms of the following series? $$3+8+22+72+266+1036\dots$$ My Approach: $$S_n=3+8+22+72+266+1036\dots T_n$$ $$-S_n=0-3-8-22-72-266-1036\dots-T_n$$ $$T_n=3+5+14+50+194+770\dots…
General Kenobi
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Sigma Notation Equation?

So I came across this post on stackoverflow which discussed the ranges of integer variables in C++. And the last point of the top-voted response was the unsigned long long int which apparently ranges from 0 to 18,446,744,073,709,551,615 (18.5…
nourgaser2012
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How can I express a sum of n variables where the output is bounded between two scalars?

How can I express a sum of n variables where the output is bounded between 0 and 10? e.g. x = n1 + n2 + n3 = 12 12 > 10 = 10
Daniel
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What's the proof for this summation with dependent variables?

Given $0\leqslant i
Selfish Stoic
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How do you evaluate a summation with variables in the parameters?

my problem is $\sum_{i=n+1}^{3n} (2i-3)$ I have done a few summations in calc 2, but I do not remember what you are supposed to do when there are variables in both parameters. I remember the rules that $i = \frac{(n^2+n)}{2} $ but Im not sure if…
Dingle Box
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Tetrahedra nth formula

I just stumbled on this interesting function $$(n* \sum(n+1))/3$$ but I can't find any reference to it online for tetrahedra numbers. I also divided $$(n* \sum(n+1)) $$ by a known nth tetrahedra formula $$(n * (n + 1) * (n + 2)) / 6 $$ which gave me…
JJ-Pysquared
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If $\ U_{r} = \frac{1+\ U_{r-1}}{2}$ and $\ U_{0}=0$, Find $\lim_{n\to\infty} \sum_{r=1}^n \ U_{r}$

I am trying to understand fully how drug half-life works. So I derived this relationship: $$\ U_{r} = \frac{1+\ U_{r-1}}{2}$$ Where $\ U_{0}=0$ and r is a set of natural numbers. My issue to how to deduce a relationship for the sum to…
Obinoscopy
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Simplifying $\sum_{m=1}^{N+1}e^{-m}$

I am currently trying to simplify $$\sum_{m=1}^{N+1}e^{-m}$$ I set $n = m-1 \implies m = n + 1$ and get $$\sum_{n=0}^{N+1}e^{-(n+1)} = \sum_{n=0}^{N+1}e^{-n-1} = \frac{1}{e}\sum_{n=0}^{N+1}e^{-n}$$ Here my question is if I have to also reduce the…
NewBiR
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Given the variables of a summation symbol can you find the other variables?

For example, $$T= \sum_{i=0}^n x^i$$ Given the values of T, x, and where i begins, can a mathematical equation be written to find the value of n. If so, can you find the value of any variable in this summation given information of the other…
Mike Wilson
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Sum $\sum_{k=1}^n{k(k-1)}$

I have a series as $$n(n-1)+(n-1)(n-2)+(n-2)(n-3)+ \ldots +n\ \text{terms}$$ How to provide a summation of this series? I think it would be similar to: $$\sum_{i=0}^{N}(n-i)(n-(i+1)).$$
christopher
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Evaluating $\sum_{n=0}^\infty n^2 x^n $ by splitting it into two summations?

How to find the sum of the following summation? $$\sum_{n=0}^\infty n^2 x^n $$ I was told to split it into two summations, but I don't see any help in that. (original problem image)
user788860
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distributive law with summations

The following is from Knuth's "Concrete Mathematics", $$\begin{align}\sum_{1 \le j, k \le 3} a_j b_k = a_1 b_1 + &a_1 b_2 + a_1 b_3 \\ + a_2 &b_1 + a_2 b_2 + a_2 b_3 \\ + a_3 &b_1 + a_3 b_2 + a_3 b_3\end{align}$$ Would it be possible to use the…
Katie Melosto
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Evaluating this finite sum help

Can someone please explain what I am missing here? Let $\mathbb{E}[X_iX_j] = \sigma_w^2 + \sigma_y^2$. $$\mathbb{E}[\sum_{i=1}^n X_i^2] = \mathbb{E}[(X_1+X_2+\cdots+X_n)(X_1+X_2+\cdots+X_n)] $$ $$\mathbb{E}[\sum_{i=1}^nX_i^2 + \sum_{i \neq j}X_i…
codenoob
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How to evaluate the sum $\sum _{k=i+j-1}^n\:1$

How would I go about solving this summation: $$\sum_{k=i+j-1}^n\:1$$
Fakhr Ali
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