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.

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What's wrong with expanding a sum this way?

$\sum_\limits{x=0}^∞ cos(x)$ = Re( $\sum_\limits{x=0}^∞ exp(ix)$) =Re ($\frac{1}{1 - \exp(ix)}$) = $\frac{1}{1 - \cos(x)}$ I know this is wrong but how would I use this technique to calculate that sum?
jdhokia
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Prove that $\sum\limits_{k=0}^n (-1)^k\:{\binom n k}^{-1}=\frac{n+1}{n+2} (1+(-1)^n)$

$$\sum _{k=0}^n (-1)^k \frac{1}{\binom n k}=\frac{n+1}{n+2} (1+(-1)^n)$$ $$A(n,k)=(-1)^k {\binom n k}^{-1}=(-1)^k \frac{(n-k)!k!}{n!}$$ $$A(n+1,k+1)-A(n+1,k)=-\frac{n+2}{n+1} A (n,k)$$ $$\sum_{k=1}^n A(n,k)=-\frac{n+1}{n+2} (A(n+1,n+1)-A(n+1,0)) =…
user215420
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the summation of multiple sums

I have a problem where I need to formalize the following summation $\sum_{i=1}^k ni $ + $\sum_{i=1}^k ni $ + $\sum_{i=1}^k ni$ + ... the sum will be repeted m times and the k values will be different in each summation notation.
aram
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Is there a closed form expression for these combinatorial sums?

I needed to compute this sum: $$\sum_{i=0}^n\binom{n}{i}i^2$$ I don't want any proofs or formulae, but just a yes/no answer. Is there a closed form expression for this sum? Just to share, I was able to find a closed form expression for this…
codetalker
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How to simplify summation with binomial coefficients

Is it possible to simplify this sum further without calculating it ?: $$ \sum_{t = 0}^{4}\left[1 - \sum_{a = 0}^{t}{4 \choose a}\left(1 \over 2\right)^{4}\right] $$ Normally I would use the binomial theorem on something similar to the inside sum but…
b_pcakes
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Sum of fourth powers in terms of sum of squares

The sum of the fourth powers of the first $n$ integers can be expressed as a multiple of the sum of squares of the first $n$ integers, i.e. $$\begin{align} \sum_{r=1}^n r^4&=\frac {n(n+1)(2n+1)(3n^3+3n-1)}{30}\\ &=\frac{3n^2+3n-1}5\cdot \frac…
Hypergeometricx
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Sum of $x_1^k+x_2^k+\dots+x_n^k$

I was recently wondering if there is some quicker way to compute $x_1^k+x_2^k+\dots+x_n^k$ for any natural $k$ than just exponentiation and adding one-by-one? Thanks in advance.
Mateusz Duchalski
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proof of $1^4+2^4+...+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$

I want a 'simple' proof to show that: $$1^4+2^4+...+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$$ I tried to prove it like the others but I can't and now I really need the proof. Also I want a geometric proof for that and this one…
Taha Akbari
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Hard summation involving binomial and quadratic

What is $$\sum \frac{2r^2-98r+1}{(100-r)({100\choose r})}$$ Where $r\in [1,99]$I have reduced it to $$\frac{(2r^2-98r+1)}{(100){99\choose r}}$$ what to do further? Partial fractions don't seem to help.
Archis Welankar
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Summation with arithmetic series

I have doubts how to solve summation if on the top there is something else than $n$ more specificially: $$\sum\limits_{i=1}^{n+1} i=\frac{(n+1)(n+2)}{2}$$ Is my solution correct? I just replace $n+1$ to $n$ in the formula for the special case of…
ocram
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Cascading Summation (2) $\sum_{i_1\le i_2\le i_3\le \cdots \le i_m}^n \left[\prod_{r=1}^m (i_r+2r-2)\right]/(2m-1)!!$

Evaluate the following summation: $$\large\sum_{i_1\le i_2\le i_3\le \cdots \le i_m}^n \left[\prod_{r=1}^m (i_r+2r-2)\right]\bigg /(2m-1)!!\\ =\large\sum_{i_1=1}^n\sum_{i_2=i_1}^n\sum_{i_3=i_2}^n\cdots\sum_{i_m=i_{m-1}}^n\frac…
Hypergeometricx
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Cascading Summation $\sum_{i=1}^n\sum_{j=i}^n\sum_{k=j}^n \frac {i(j+2)(k+4)}{15} $

Evaluate $$\sum_{i=1}^n\sum_{j=i}^n\sum_{k=j}^n \frac {i(j+2)(k+4)}{15} $$ Background Many basic summation questions on MSE relate to a single index - it might be interesting to devise a question where the summand is a product of the three indices…
Hypergeometricx
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What is the expanded form of $\sum_\limits{0}^{0}{f(x)}$?

The is a MCQ in my math book which says the following: Expanded form of $\sum\limits_0^0{f(x)}$ is: 1) $0$ 2) $f(0)$ 3) $1$ 4) None I don't know which one is correct but one of the first two is correct.
user41736
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Converting Σ i(i + 1) into a formula, given this hint

The given summation is: $$\sum_{i=1}^n i(i+1)$$ The goal is to convert it into a formula which only uses n. Solving this, I got the answer: $$\frac{n}{3}(n+1)(n+2)$$ However, I don't believe the way I solved it is the intended method, despite…
C T
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Calculating the 12 days of christmas by hand

For an exercise in my math class we are calculating the cost of the 12 days of christmas. Let's define a set $c$ to be the price of each item in the popular "12 days of christmas" song, from a partridge in a pear tree all the way up to the drummers…
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