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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How can I calculate summated math questions without using some kind of computer script or program?

For example, how can I find the sum of $\frac 1 {(2 + n) \cdot (3 + n)}$ + ... for $n = 0$ to $n = 97$ i.e. $$\frac{1}{2 \cdot 3} + \frac 1 {3 \cdot 4} + ... + \frac 1 {99 + 100}$$
Jason
  • 3,563
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Cardinality summation

How do I prove that, for set $X$, $$\sum_{S\subseteq X, S\neq \emptyset}\frac{(-1)^{|S|}}{|X|+|S|} = \frac{|X|!(|X|-1)!}{(2|X|)!}$$ I have been around this exercise all day and would much appreciate your help.
Maria
  • 171
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What is $\sum_{k=0}^n {n \choose k} p^k (1-p)^{n-k} e^{-s(k - np)^2}$?

Consider the following expression in three variables, $0 \leq p,s \leq 1$ and $n >0$ $$S_{n, p, s} = \sum_{k=0}^n {n \choose k} p^k (1-p)^{n-k} e^{-s(k - np)^2}$$ If $s = 0$ then $S_{n, p, 0} = 1$. Is there a closed form for the sum for $ s > 0$? …
user35671
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I have a huge expression that I have evidence that it can be simplified but don't know how

During work on my thesis, I am working on formulating a solution to some problem and I came across a huge and very complicated formula. I tried plugging in a lot of different values and I have evidence that it can be simplified. The formula is as…
Ran Elgiser
  • 63
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Showing that $\sum^{n}_{k=1}\frac{(-1)^{k}}{k} = -\log(2)+(-1)^n\cdot \int^{t=0}_{t=1}\frac{t^n}{(1+t)}$

I'm supposed to prove the following: $$\sum^{n}_{k=1}\frac{(-1)^{k}}{k} = -\log(2)+(-1)^n\int_0^1\frac{t^n}{t+1}\ \mathrm dt$$ Defining $$U(n)=\int_0^1\frac{t^n}{t+1}\ \mathrm dt$$ I found that $$U(n)+U(n-1)=\frac1n$$ We are supposed to use the…
Vevitek
  • 63
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Evaluate $\sum_{r=2}^{\infty} \frac{2-r}{r(r+1)(r+2)}$

Evaluate $$\sum_{r=2}^{\infty} \frac{2-r}{r(r+1)(r+2)}$$ So in a previous part of the question I calculated that $$\sum_{r=1}^{n} \frac{2-r}{r(r+1)(r+2)} = \sum_{r=1}^{n}\left(…
H.Linkhorn
  • 1,283
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How to rewrite a Sum?

How to find the following sum? $$ \frac{2}{n} \sum_{i=1}^{n} \left(\frac{4i}{n} +1\right) $$ For example $$ \sum_{i = 1}^n i^2 = \frac{n(n+1)(2n+1)}{6}. $$ I need to find the value of which this would converge to and I know that it must be six…
J-Dorman
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Sum of the cubes of the first $n$ natural numbers

I know that the sum of the cubes of the first $n$ natural numbers is $\{\frac {n(n+1)}2\}^2$ but I am looking for a method to derive this. If there is a method please try to make as simple as possible and possibly without the use of mathematical…
Prakhar Nagpal
  • 1,312
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Calculate the sum of series $\sum\limits_{i=0}^{n-1} i2^i$

Possible Duplicate: How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$? How can I calculate precise value of that series: $\sum\limits_{i=0}^{n-1} i2^i$ ? So far, I tried to differentiate the $ \sum\limits_{i=0}^{n} 2^i = 2^{i-1} - 1…
Greg Witczak
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Summation of product of three terms in AP

Here is an interesting relationship: $$\sum_{r=1}^n r(r+m)(r+2m)=\left(\sum_{r=1}^n r\right)\left(\sum_{r=1}^{n+2m} r\right)\tag{1}$$ $$\text{i.e.} \;\; 1\cdot(1+m)(1+2m)+2\cdot (2+m)(2+2m)+3\cdot (3+m)(3+2m)+\cdots…
Hypergeometricx
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Prove that sum is an integer.

Prove that: $\sqrt{2-\frac{1}{1^2+\sqrt{1^4+\frac{1}{4}}}}+\sqrt{2-\frac{1}{2^2+\sqrt{2^4+\frac{1}{4}}}}+\sqrt{2-\frac{1}{3^2+\sqrt{3^4+\frac{1}{4}}}}+\cdots+\sqrt{2-\frac{1}{119^2+\sqrt{119^4+\frac{1}{4}}}}$ is an integer. I noticed that…
The Math Troll
  • 383
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sum of the series?

let sum of the series $$S=1-1+1-1+1-1+1-1+1-1........$$ $S=\frac{1}{2}$ my question is if there are even number of terms then the sum is $0$ and if the number of terms are odd then sum is $1$. but we don't know whether its odd or even because the…
Nebo Alex
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How to show $\sum_{k=1}^n \binom{n-1}{k-1} n^{n-k} k! = n^n$

To me it appears that lhs is some fancy way to count all functions from $[n]$ to $[n]$. I tried several approaches, including interpreting $\binom{n-1}{k-1}$ as number of solutions for $n=x_1+\dots +x_k$ where $\forall_i x_i \ge 1$. But it didn't…
Krzysztof Lewko
  • 2,312
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How to convert sum into formula?

I am reading some combinatorics books. And here author first obtained a sum answer for a problem and then converted it to formula without explaining it. Just writing equality sign. This is that expression: $$2(2\sum_{k=1}^{n-1} k(k-1) + n(n-1)) =…
Turkhan Badalov
  • 1,083
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Closed form of the summation- $\sum_{r=1}^{n}\frac{r^24^r}{(r+1)(r+2)}$

I have got the following summation-$$\displaystyle\sum_{r=1}^{n}\frac{r^24^r}{(r+1)(r+2)}.$$ I have to find the closed form or the general form to find the sum of this series. I know upto summation of Telescopic Series and Some special series like…
Soham
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