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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Show that $ \sum\limits_{k=1}^\infty \frac{3^k-2^k}{k\cdot6^k}=\ln\frac43$

$\sum_{k=1}^\infty \frac{3^k-2^k}{k\cdot6^k}$ i must prove this sum converges to $\ln(4/3)$. i tried to write expresion :$\frac{3^k-2^k}{k\cdot6^k}=\frac 1 k \left(\frac 1 {2^k-3^k}\right)$ and make two sums but i tink this sums is a dezvoltation…
Ica Sandu
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How do you solve this summation: $\displaystyle\sum_{n=1}^{\infty} \dfrac{n^3}{2^n}$?

How would you solve $\displaystyle\sum_{n=1}^{\infty} \dfrac{n^3}{2^n}$? I stumbled upon it online and it has been giving me a very difficult time despite how simple it looks. I would love it if someone could explain it to me.
Dustin Luong
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Calculate $\sum\limits_{n\geq 1}\frac{1}{1+2+\ldots+n} $

Calculate $\displaystyle \sum_{n\geq 1}\dfrac{1}{1+2+\ldots+n} $ My attempts : \begin{aligned} \sum_{n\geq 1}\dfrac{1}{1+2+\ldots+n} &= \sum_{n\geq 1}\dfrac{2}{n(n+1)}=2\sum_{n\geq 1}\left( \dfrac{1}{n}-\dfrac{1}{n+1}\right)=2\left(…
Yacob
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Difference between $\sum_{n=1}^{k} f(n)+\sum_{n=1}^{k} g(n)$ and $\sum_{n=1}^{k} [f(n)+g(n)] $

What is difference between $$\sum_{n=1}^{k} f(n)+\sum_{n=1}^{k} g(n)$$ and $$\sum_{n=1}^{k} [f(n)+g(n)] $$
Soru
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Summation of $\frac{1}{4k+3}, \frac{1}{4k-1}, \frac{1}{2k-1}$ and $\frac{1}{2k+1}$

Question: How do I show that $$\varphi(2,n)-\varphi(4,n)=2\sum\limits_{k=1}^n\frac 1{\{2(2k-1)\}^3-2(2k-1)}$$ Where$$\varphi(2,n)=1+2\sum\limits_{k=1}^n\frac 1{(2k)^3-2k}$$$$\varphi(4,n)=1+2\sum\limits_{k=1}^n\frac 1{(4k)^3-4k}$$ I started with…
Crescendo
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Show that $(2n+1)$ is a factor of the sum of even powers of the first $n$ integers

We can easily show that $n$ is a factor of the sum of $p$-th powers $(p\in\mathbb N)$ of the first $n$ integers , by assuming that the sum is a general polynomial of order $p+1$, and setting $n=0$, giving a zero constant term (as the sum is the same…
Hypergeometricx
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Given that $\sum^{n}_{r=-2}{r^3}$ can be written in the form $an^4+bn^3+cn^2+dn+e$, show that: $\sum^{n}_{r=0}{r^3}=\frac14n^2(n+1)^2$

Question: Given that $\displaystyle\sum^{n}_{r=-2}{r^3}$ can be written in the form $an^4+bn^3+cn^2+dn+e$, show that: $$\sum^{n}_{r=0}{r^3}=\frac14n^2(n+1)^2$$ Attempt: Substituting $n = -2,-1,0,1,2$into $\sum_{r=-1}^{n}{r^3}$ we…
Sonny Da Silva-Peters
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Is $\sum_{n=0}^{\infty} \frac{n^2}{(b-n^2)(a-n^2)}$ expressible in terms of trigonometric functions

I recently ran into the sum $$S=\sum_{n=0}^{\infty} \frac{n^2}{(\alpha-n^2)(\beta-n^2)}.$$ Mathematica gives it in terms of the Digamma function as $$S=\frac{-\alpha \psi ^{(0)}(1-\alpha )+\alpha \psi ^{(0)}(\alpha +1)+\beta (\psi ^{(0)}(1-\beta…
Wolpertinger
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Evaluate $\sum{\left(\frac{1}{1+a_n}\right)}$ where $a_n=(-1)^{n+1}\sum{\left(\binom{2n+1}{2k}2^k\right)}$

Suppose $\left\{a_n\right\}$ is a sequence defined by: $$a_n:=\left(-1\right)^{n+1} \sum_{k=0}^n {\left(\binom{2n+1}{2k}2^k\right)}$$ The problem is to evaluate: $$\sum_{n=1}^{\infty} {\left(\frac{1}{1+a_n}\right)}$$ An estimate for the value of the…
Ant
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An Elementary Summation Problem with $\varphi(4,n)$

The complete proof is given by Ramanujan.$$\begin{align*}\varphi(4,n) & =1+\sum\limits_{k=1}^n\left\{\frac 1{4k-1}+\frac 1{4k+1}-\frac 1{2k}\right\}\\ & =\sum\limits_{k=1}^{4n+1}\frac 1k-\frac 12\sum\limits_{k=1}^{2n}\frac 1k-\frac…
Crescendo
  • 4,089
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How to delineate a summation of digits in a number?

Let's say we have an integer x. How should I delineate a summation of its' digits until I get a single digit. For example: x = 12345 1 + 2 + 3 + 4 + 5 = 15 1 + 5 = 6 I believe it is possible, just can't find a mathematical way to do so. Would be…
user48512
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Why $\sum_{i=0}^\infty i^2p^{i-1} = \sum_{i=0}^\infty i(\frac{d}{dp}p^{i-1})$?

I'm going through a proof right now and am having trouble figuring out the math behind one line. It says: $$\sum_{i=0}^\infty i^2p^{i-1} = \sum_{i=0}^\infty i(\frac{d}{dp}p^i) $$ I know this question is vague but can anybody explain why this is the…
MarksCode
  • 169
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Exponential Backoff Equations

I'm admittedly not great at math, and I'm trying to work through the wikipedia article on exponential backoff, and there are a few things I don't fully understand. The article says: Given a uniform distribution of backoff times, the expected backoff…
tmountain
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How can I solve this sum? $\sum_{k=1}^{n}k2^{k-1}$

How can I solve this sum? $\sum_{k=1}^{n}k2^{k-1}$ Is $\sum_{k=1}^{n}k2^{k-1}$ equals to $\sum_{k=1}^{n}k * \sum_{k=1}^{n}2^{k-1}$
Andrey França
  • 159
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Sum order index

In the following sum, I encounter an issue whereby m ranges from 1 to 0 - when k = j (which is guaranteed for every p). Is this a sensible sum or do I need to make changes? The 'j-k' stems from the following sum: It is a binomial expansion…
OverLordGoldDragon
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