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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Calculate sum of squares of first $n$ odd numbers

Is there an analytical expression for the summation $$1^2+3^2+5^2+\cdots+(2n-1)^2,$$ and how do you derive it?
LWZ
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Moving index in a summation - how does that work?

In a book I have they solve an equation like this: $$\sum_{i=2}^{19} i(i - 2) = \sum_{i=2 - 1}^{19 - 1} (i + 1)(i + 1 - 2) = \sum_{i=1}^{18} (i + 1)(i - 1) = \cdots$$ What I don't get is how it works. If I do $-1$ on the summation index, why do I…
BMBM
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Evaluate $\sum_{n=1}^{\infty} {\left(\frac{(-1)^n}{n}\sum_{k=1}^n {\left(\frac{(-1)^k}{k}\right)}\right)}$

Evaluate: $$\sum_{n=1}^{\infty} {\left(\frac{(-1)^n}{n}\sum_{k=1}^n {\left(\frac{(-1)^k}{k}\right)}\right)}$$ It looks like the series for $\ln(2)$ 'embedded in itself', so my guess for the value is $\ln^2(2)$. Unfortunately, this is not correct, as…
Ant
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Evaluating the following summation

$\require{cancel}$ Evaluate $$\sum_{n=2}^{\infty}\ln\left(\frac{n^2-1}{n^2}\right)$$ My procedure: $$\sum_{n=2}^{\infty}\ln\left(\frac{n^2-1}{n^2}\right)=\sum_{n=2}^{\infty}(\ln\left(n-1\right)-\ln(n)+\ln(n+1)-\ln(n))$$ If we evaluate a few…
DMH16
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How to notate a While loop?

I've noticed that Sigma notation is a lot like a For loop in programming. What, if anything, can be used to notate a While loop mathematically? In other words, how to you notate the sum of everything (like Sigma) while a variable is equal or…
Nick Suwyn
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How find this sum $\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$

Question: Find the sum $$I=\sum_{n=1}^{\infty}\dfrac{L_{n}(2)}{n^4}=?$$ where …
math110
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No real solutions to $\sum_{n\,=\,0}^\infty \frac {(-1)^{n + 1}} {n^x} = 0$

Prove that there are no real numbers $x$ such that $$\sum_{n\,=\,0}^\infty \frac {(-1)^{n + 1}} {n^x} = 0$$ Can I have a hint please?
Superbus
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Error in Gradsteyn and Ryzhik?

I am looking in Gradsteyn and Ryzhik 7th edition and formula 0.241.11 is the following finite sum $$\sum_{k = 0}^i{\binom{i + k}{k}^{i - k}k} = (i + 1)4^i - (2i + 1)\binom{2i}{i}.$$ However, after looking at a few values of $i$ it appears that this…
Paul
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Convert Summation into Formula

I have the formula for finding the average time of an linear search algorithm as below \begin{equation*} AT\ =\frac{\ \sum ^{n+1}_{i\ =\ 1} \ \theta ( i)}{n+1}. \end{equation*} The above formula was simplified to become the one below, I am not sure…
George
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about the following sum : $\frac{1}{1 \cdot 3}+\frac{2}{1 \cdot 3 \cdot5}++\ldots+\frac{n}{1 \cdot 3 \cdot 5 \cdot7 \cdot \ldots \cdot (2n+1)}$

Any hint about this expression : $$\frac{1}{1 \cdot 3}+\frac{2}{1 \cdot 3 \cdot5}+\frac{3}{1 \cdot3 \cdot5 \cdot7}+\ldots+\frac{n}{1 \cdot 3 \cdot 5 \cdot7 \cdot \ldots \cdot (2n+1)}$$ Thanks :) there must be a trick :)
Iuli
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Sum of the series $\sum_{n=1}^\infty n^2e^{-n}$

The question is to find out the sum of the series $$\sum_{n=1}^\infty n^2 e^{-n}$$ I tried to bring the summation in some form of telescoping series but failed. I then tried approximating the sum by the corresponding integral(which I am not sure…
Navin
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Equality of the sums $\sum\limits_{v=0}^k \frac{k^v}{v!}$ and $\sum\limits_{v=0}^k \frac{v^v (k-v)^{k-v}}{v!(k-v)!}$

How can one proof the equality $$\sum\limits_{v=0}^k \frac{k^v}{v!}=\sum\limits_{v=0}^k \frac{v^v (k-v)^{k-v}}{v!(k-v)!}$$ for $k\in\mathbb{N}_0$? Induction and generating functions don't seem to be useful. The generation function of the right sum…
user90369
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Find $\sum_{m=0}^n\ (-1)^m m^n {n \choose m}$

I'm going to university in October and thought I'd have a go at a few questions from one of their past papers. I have completed the majority of this question but I'm stuck on the very last part. In honesty I've been working on this paper a while now…
Aka_aka_aka_ak
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Is the sum of reciprocals of all products from $2$ to $n-1$ always $0.5n-1$?

I was looking up riddles for my math classes to work on for the end of the year and found the following riddle. http://mathriddles.williams.edu/?p=129 I followed the advice and started working with examples of small numbers and stumbled upon a…
Dirigible
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Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction?

I oversaw a high-school mathematics test the other day, and one of the problems was the following Show, using induction or other means, that $$\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$$ The induction proof is very standard, where the induction…
Arthur
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