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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Triple finite sum

$\displaystyle \sum_{i=1}^a \sum_{j=1}^b \sum_{k=1}^c f(i,j,k)$ where a,b,c are fixed natural numbers and assuming $f(i,j,k)=i+j+k$. How do we calculate that sum? I mean is there any type for that sum? Function $f$ includes $i, j$ and $k$ and that …
Vasilis Manavis
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Differentiation of summation

How do I show that \begin{align} \sum_{n=1}^\infty nz^{n-1} = \sum_{n=0}^{\infty} (n+1)z^n \end{align} (This is a result of differentiating $\sum_{n=1}^{\infty} z^n$ with respect to the variable $z$.)
Cookie
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Summation of Number of Digits

I have seen this problem a while ago and I wasn't able to find the answer. $$ \begin{align*} P_n & = nd(7^n) \\ S & = \sum_{n=1}^{3981} P_n \end{align*} $$ Where $nd(n)$ is the number of digits of $n$, such that: $$ \begin{align*} nd&(5) & = 1…
N1xx1
  • 121
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Is this summation $> 0$ or $< 0$?

Sum: $$ s =\frac{\left(\sum_{i=1}^n a_i^{p-1} v_i\right)\left(\sum_{i=1}^{n} a_i^{p-1} v_i\right)}{\sum_{i=1}^{n} a_i^p} - \sum_{i=1}^{n} a_i^{p-2} v_i^2 $$ Where $0 0
JDS
  • 713
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Is there a formula for calculating the sum over squares?

I know that there are a lot of formulas about sums. However, I would like to ask, is there any formula that calculate the sum over squares? What I mean is that I want a formula if that exists, which gives answer to the following: $$1+2^2 + 3^2 + 4^2…
K. Stasko
  • 657
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Is there a Relationship between sum and sum of reciprocals?

Is there any relationship between the sum of positive numbers and the sum of the reciprocals of the same numbers? For example: $A=1+2+3.5$ and $B=1+1/2+ 10/35$ Thanks!
Paulo
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Evaluate: $\sum _{k=2}^{n}\frac{n!}{(n-k)!(k-2)!}$.

Let $n ≥ 2$. Evaluate: $$\sum_{k=2}^{n}\frac{n!}{(n-k)!(k-2)!}$$ unable to solve this series problem.
ghugni
  • 99
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Evaluating $\sum\limits_{x=1}^{2012} \frac{9^\frac{x}{2013}}{9^\frac{x}{2013}+3}$

$\Large \sum\limits_{x=1}^{2012} \frac{9^\frac{x}{2013}}{9^\frac{x}{2013}+3}=?$ I thought that if $\Large\frac{9^k}{9^k+3}$ was equal to $\Large\frac{9^{1-k}}{9^{1-k}+3}$ it would be easy to solve but apparently it isn't.
Zafer Cesur
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Convert 2 + 5 +8+ . . . + (6n-1) = n(6n+1) to Sigma Notation

2 + 5 + 8 + . . . + (6n-1) = n(6n+1) This is what I have so far. The sum of (3j-1) from j=1 to something I`m not sure of.
user114027
  • 95
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Summation of $\sum\limits_{j=2}^n j = \frac{n(n+1)}{2} -1$

I've know that $\sum\limits_{i=1}^n k$ is $ \frac{n(n+1)}{2}$. Then why $\sum\limits_{j=2}^n j = \frac{n(n+1)}{2} -1$ ? I understand that i=1 $\neq$ j=2 is the key, but I can't get further. from : Introduction to Algorithms (3rd edition) p. 27
user112780
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How to correctly represent a nested sum

Suppose I have a matrix: $$ A = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \\ \end{pmatrix} $$ For which i want to sum the elements. My first inclination is to write: $$ \sum_j \sum_i A_{i,j} $$ But I…
Manetheran
  • 123
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Is there a closed form for $\sum_{k_1 + \cdots+k_m = n} \frac{\Gamma(k_1 + \alpha)}{k_1!} \ldots \frac{\Gamma(k_m + \alpha)}{k_m!}$?

Let $\alpha$ be a positive real number, and $n,m$ two positive integers. Is there a closed form for this sum? $$\sum_{k_1 + \cdots+k_m = n} \frac{\Gamma(k_1 + \alpha)}{k_1!} \ldots \frac{\Gamma(k_m + \alpha)}{k_m!}$$ where the sum goes through all…
a06e
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Sigma notation question

Find the value of the sum. $$\sum_{i=1}^n i(i+1)(i+2)$$ Does this mean that the answer is $$1(1+1)(1+2) + \cdots + n(n+1)(n+2)$$? Is there no value to the answer?
Daniel Valentine
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Evaluating $\sum_{n=0}^\infty \frac{n}{3^n}$

How can i prove that $ \sum\limits_{n=0}^\infty \frac{n}{3^n}=\frac{3}{4}$? I tried to find some pattern but didn't succeed
mori
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How to work out a double summation without manually finding each answer?

I have the double summation $$ \sum_{p = 1}^{2} \sum_{q = p}^{3} {(p-1)q} $$ I know how to work this out if I was to go through and add up every term. My question is, how would I do this in a shorter way, so if the 2 and the 3 were 200 and 300, I…
Maria
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