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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Sum of Series Involving Factorials

The problem is to prove that : $\frac{2}{0!+1!+2!} + \frac{3}{1!+2!+3!} + \dots + \frac{n}{(n-2)!+(n-1)!+n!} = 1 - \frac{1}{n!}$ I used the following rather lengthy approach to prove this: The general term $\frac{k}{(k-2)!+(k-1)!+k!}$ can be…
erdoswiles
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Why is $\sum_{m>n}p(1-p)^{m-1}$ equal to $(1-p)^n \sum_{m>0}p(1-p)^{m-1}$?

Suppose $p \in (0,1)$, $n,m \in \mathbb{N}$, we have: $$\sum_{m>n}p(1-p)^{m-1} = (1-p)^n \sum_{m>0}p(1-p)^{m-1} $$ Why is this true? I can't seem to find the reasoning behind it.
user401855
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Question on a tricky Arithmo-Geometric Progression::

$$\dfrac{1}{4}+\dfrac{2}{8}+\dfrac{3}{16}+\dfrac{4}{32}+\dfrac{5}{64}+\cdots\infty$$ This summation was irritating me from the start,I don't know how to attempt this ,tried unsuccessful attempts though.
satyatech
  • 517
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Summation involving the integer nearest to $\sqrt n$.

Let $\langle n\rangle$ denote the integer nearest to $\sqrt n$. Evaluate $$\sum_{n=1}^{\infty} \frac{2^{\langle n\rangle}+2^{-\langle n\rangle}}{2^n}.$$ I tried writing down a few terms but I couldn't get any idea of how the sum progress. Any…
Navin
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Solve geometric series?

I think this is a geometric series, Im in way over my head here and not even sure of the vocab or how to ask really. I have $$\sum_{1}^n P_0(1+r)^n$$ And when $r=.1$ and $P_0=1$ I would like to know how to find $n$ when the $\sum_{1}^n…
ARs
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How do I write these in summation/product notation?

I have difficulties in writing some equations in summation/product notation. I want to write this in summation notation. $$p_1p_2+p_1p_3+...+p_1p_n+p_2p_3+p_2p_4+...+p_2p_n+...+p_{n-1}p_n$$ is it $$\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}p_ip_j \quad…
fairytale
  • 151
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Find cumulative days in a date

Say you have a date, 30/03/2017, and you want to find out how many days are within that date from 00/00/0000. I know the days portion and year portion of the date can be expressed as: cumulativeDays = year * 365 + year / 4 + days; However, I don't…
goastler
  • 121
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Summation by parts (finite calculus) of $\sum\limits_{i=0}^{n}i\cdot a^i$

I'm studying the calculus of finite differences and have read about summation by parts: $\sum f(x)\Delta g(x) = f(x)g(x) - \sum g(x+h)\Delta f(x)$. The tutorials I'm using go a bit woolly at the point of introducing this technique: they use examples…
Adam
  • 97
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Determine: $S = \frac{1}{2}{n \choose 0} + \frac{1}{3}{n \choose 1} + \cdots + \frac{1}{n+2}{n \choose n}$

I am studying the book Equations and Inequalities by Herman et al, and am stuck on the following exercise: Determine: $S = \frac{1}{2}{n \choose 0} + \frac{1}{3}{n \choose 1} + \cdots + \frac{1}{n+2}{n \choose n}$ The hint says to consider…
Adam
  • 97
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proof by by directly manipulating the sum:

Prove that $$\sum_{i=0}^{n} x ^{i} = \frac{ 1-x ^{n+1} }{ 1-x }$$ to be used by directly manipulating the sum: let A be the sum, and show that xA = A + x^(n+1) -1 I don't get how its going to equal $\frac{ 1-x ^{n+1} }{ 1-x }$ $$xA=x\sum_{i=0}^n…
Jack F
  • 349
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Transformed Sum question

I am trying to understand how this sum was transformed from $$\sum_{n=1}^\infty \frac {\sqrt{n}}{n(n+1)}$$ to $$ 1 + \sum_{n=2}^\infty \frac{\sqrt{n}-\sqrt{n-1}}{n} $$ I see that the index was changed from $n=1$ to $n=2$, thus requiring that…
john fowles
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The solution to a summation $4^n/(4^n+2 )$

I found this problem on summations, and I'm not really sure how to solve it. Could someone give a hint as to how to do so? Find the value of $$\sum_{i=1}^{1000}f\left(\frac{i}{1000}\right),\qquad f(x) = \frac{4^x}{4^x+2}$$ It came on an exam where…
sleepyfox128
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Some of series involving factorial in the denominator

What would be the sum of the series $\dfrac{n^2}{n!}$ ? I don't even know where to start with. It's nothing like telescopic. I tried to compare with some known series but that doesn't seems to work.
Nitish
  • 801
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How to compute sums like $\sum_{n=1}^{\infty} \frac{n^a}{b^n} $

I have many sums to compute but all of them are similars to each other. There are of this kind: $$\sum_{n=1}^{\infty} \frac{n^a}{b^n}$$ $$a,b \in \mathbb{Z}$$ $$a \in [1,\infty)$$ $$b \in (a,\infty)$$ For instance: $$\sum_{n=1}^{\infty}…
Piotr Wasilewicz
  • 306
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Prove the following identity without using induction: $\sum\limits_{i=0}^n a^i = \frac{1-a^{n+1}}{1-a}$ for $a\neq1$

I'm struggling with proving $$ \sum^n_{i=0}a^i = \frac{1-a^{n+1}}{1-a} $$ for $a\neq1$, not sure where to start.
mirai
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