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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Constants in the sigma notation for sums

This really is a question about the semantics of the sigma notation for writing long sums in a concise way. Let's say we have a sum given by the following notation, $$\sum_{i = 0}^{n} (\frac{1}{n} + i^2)$$ My question now is rather simple. For a…
roblox99
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Why is Riemann's Rearrangement Theorem not considered a contradiction?

If we can pick any conditionally convergent series, $\sum_{n=1}^\infty a_n$, and pick any real number, $\ell$, and show that $$\sum_{n=1}^\infty a_n = \ell,$$ then why is that not considered a contradiction, since we can show that this sum takes on…
Aaron Kirk
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Let $a_n=|nx-1|$, $n\in\mathbb{Z}^+$ and $x\in\mathbb{R}$. Find the least possible value of $\sum_{n=1}^{696}a_n$

Let $a_n=|nx-1|$, $n\in\mathbb{Z}^+$ and $x\in\mathbb{R}$. Find the least possible value of $\displaystyle\sum_{n=1}^{696}a_n$ My attempt: \begin{align*} S_{696}&=\sum_{n=1}^{696}|nx-1|\\ &=|x|\sum_{n=1}^{696}|n-\frac{1}{x}|\\ \end{align*}
Ken
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The sum $\sum_{k=1}^{n-1}{10k+5\over (k+1)!2^k}$

I'm solving a problem, and I'm getting the following sum as the solution. $$\displaystyle\sum_{k=1}^{n-1}{10k+5\over (k+1)!2^k}$$ Wolfram Alpha says this can be simplified. How do I arrive at the result? I believe I've calculated similar sums in the…
Bartek
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Multiplication of two polynomials

Let us say we are given two polynomials $p=\sum_{k=0}^{m}p_kA^k$ and $q=\sum_{j}^{l}q_jA^j$ where we say that for $mm$. The equation goes as…
babemcnuggets
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How many tuples $( A , B , C ) $ of positive integers satisfy $A × B + C = N$

This is a programming question I found on Atcoder and I am trying to solve it just using mathematics. I want a simple formula to find number of tuples without using a computer. So I derived a formula to calculate it: $Sn = \sum_{x=1}^n…
Mohammad Alshareef
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Hint for finding the sum: $\sum_{n=0}^{\infty}\frac{1}{4n^2-1} x^{2n+1}$?

I know that $\sum_{n=0}^{\infty}\frac{1}{4n^2-1}$ is a telescoping sum, but $x^{2n+1}$ in the sum complicates it a bit.
Labbsserts
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$\sum\limits_{k=1}^n \frac{x^k}{k}=H_n+\sum\limits_{k=1}^n \binom{n}{k} \frac{(x-1)^k}{k}$

Prove: $$\sum\limits_{k=1}^n \frac{x^k}{k}=H_n+\sum\limits_{k=1}^n \binom{n}{k} \frac{(x-1)^k}{k}$$ where $H_n=\sum\limits_{k=1}^n\frac1k$ For the $x^k$ i tried $x^k=((x-1)+1)^k$ and decompose it into binomial expansion but I got dual sum at the…
Jale'de jaled
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how many iterations would there be through a double sum?

$\sum^{100}_{i=1}\sum_{j \neq i} A_iB_j$ How many total summations would happen within this expression? I'm thinking 100*99? Since there are 100 on the outer sum, and then one less on the inner sum? Thanks
Katie Melosto
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Prove that $\sum_{i \in A} i k_i = \sum_{i \in B} i \Rightarrow \sum_{i \in A} i^2 k_i \neq \sum_{i \in B} i^2$

I would like to prove that, given two disjoint sets of integers A and B (with elements strictly greater than 1), and a sequence of integers $k_i > 0$, then $\sum_{i \in A} i k_i = \sum_{i \in B} i \Rightarrow \sum_{i \in A} i^2 k_i \neq \sum_{i \in…
weghiuop
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Prove that exists a function $f$, such that $\sum_{i=0}^{n} i (c_i - 1) = 0 \Leftrightarrow \sum_{i=0}^{n} f(i) (c_i - 1) \neq 0$

I would like to prove that exists a function $f$ such that $\sum_{i=0}^{n} i (c_i - 1) = 0 \Leftrightarrow \sum_{i=0}^{n} f(i) (c_i - 1) \neq 0$ $\forall \: n > 0$. This must hold $\forall \: c_i \geq 0$, with $c_i$ being an integer, and with…
weghiuop
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Does this sum have an analytic form?

Does this sum have an analytic form? $$\sum_{n=1}^{\infty}\frac{a^{n}\left(-1\right)^{n}}{n\sqrt{n+1}}$$ Numerically, it appears to only converge on $-1 < a \le 1$
Jerry Guern
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How to sum $nC^{n-1}$ from 0 to infinity?

Where C is a const. e.g. sum $n(.5)^{n-1}$, OR sum $n(2)^{n-1}$ from 0 to infinity. Thank you.
RHS
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An algebra summation problem

Given 1D arrays $A$, $B$, $C$, $D$ and $E$, all of length 3. Please verify that： $$ 2 \sum_{i=1}^{3} \sum_{j=1}^{3} \sum_{k=1}^{3} \sum_{n=1}^{3} \ \epsilon_{ijk} \ A_{n} \ B_{j} \ C_{k} \ D_{n} \ E_{i} + \sum_{i=1}^{3} \sum_{j=1}^{3} \sum_{k=1}^{3}…
declmal
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Simplify $\sum_{i=1}^{n}\sum_{j=i+1}^{n} (i+j)$

$$\sum_{i=1}^n \sum_{j=i+1}^n (i+j)$$ I'm pretty sure it's somewhat simple to solve but I can't get it done. I know that the given sum is equal to $\frac{n}{2} (n^2-1)$. I've been on this for 3h+ now and I'm really hoping someone can give me a hint…
Florian
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