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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Simplification of formula with finite set partial sums

As part of a proof I am working on, I have derived the following formula: $$f(n,P)=\frac{3^{\#P}n+\sum_{a=0}^{\#P-1}3^{a}(2^{(\sum_{b=a}^{\#P-a-1}P_{b})})}{2^{\sum P}} \\\text{where }n\in2\mathbb{Z}+1,\text{ P=Ordered finite set of #P cardinality…
Ian F
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Evaluate $\sum_{n=1}^\infty 2^{-\frac{n}{2}}$

Find $$\sum_{n=1}^\infty 2^{-\frac{n}{2}}$$ I know that the final numerical value of that is $1+\sqrt2$ but not sure how to get that. Any identities, formula or hints would be helpful. I tried re-expressing it as…
qydyqydy
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Evaluate infinite sum

If $a_n = n +{1 \over n}$ then find $$\sum_{n=1}^{\infty}{(-1)^{n+1}}{a_{n} \over n!}$$ My work : $e^{-x}=1-\frac{x}{1!}+\cdots$ $$\sum (-1)^{n+1}\left(n+{1 \over n}\right)\cdot{1 \over n!}$$ $$=e^{-1}-\sum(-1)^{n+1}{1 \over {n\cdot n!}}$$ Now how…
Ronin
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How to change upper and lower limits of summation

Given a problem like this one $$\displaystyle \sum_{i=-20}^0 \left(\dfrac{1}{3}\right)^i$$ what would I have to do to the summation to make it go from 0 to 20? I'm assuming you can't just switch the limits without doing anything else to the…
jcampos12
  • 13
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exchanging order of double sums

I have the double sum $$ \sum_{n=0}^\infty \sum_{k=1}^n q(k)p(n-k) $$ It's important that $p(s)=0$ whenever $s<0$. I think it's ok to write $$ \sum_{k=1}^\infty \sum_{n=k}^\infty q(k)p(n-k), $$ but I'm unsure on this. My reasoning to get here from…
kevinkayaks
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Summation or Algebra?

I am trying to visualize the concept that should be involved with this problem: What is $x$ in the equation $$(x-1)-2(x-2)+3(x-3)-4(x-4)+\dots-10(x-10)=0\quad ?$$ What should I do? Express in summation (which I find very hard) or just algebra?
Pisting Inatay
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Could someone please show me the steps of this sum?

In my probability textbook, I saw this summation, $\sum_{n=0}^{N}$$N\choose{n}$$s^n$$=(1+s)^N$ but I have no idea why it stands, could someone please show me the steps in between? (Or link me something that is similar). Much appreciated!
UnsinkableSam
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sum from 1 to infinity (x^k)/(2k-1)

$$\sum_{k=1}^\infty {\frac{x^k}{2k-1}} $$ Is there a closed form for this infinite sum? |x|< 1 .This is very similar to the infinite sum for log(1-z) but only being divided by odd integers. Can i modify the log function to arrive at this sum through…
Kedar Yadav
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Is my summation result correct? $\sum_{i=0}^{n-1} \sum_{j=0}^{i-1} (i+j) =\cdots= \frac{n(n-1)^2}{2}$

Is this result correct? $$\begin{align} \sum_{i=0}^{n-1} \sum_{j=0}^{i-1} (i+j) &= \sum_{i=0}^{n-1}\left( i^2 + \frac{i(i-1)}{2} \right)\\ &= \frac{3}{2}\sum_{i=0}^{n-1}i^2 - \frac{1}{2}\sum_{i=0}^{n-1}i \\ &= \frac{3}{2}\frac{n(n-1)(2n-1)}{2} +…
gust
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How would I go about computing this finite sum?

How would I go about computing the sum $$ \sum_{k=1}^{n} \dfrac{(-k^2+2k+1)2^k}{(k(k+1))^2}. $$ I have tried partial fractions but have gotten stuck trying to find the coefficients. I decomposed it like this: $$ \dfrac{2^k(-k^2+2k+1)}{(k(k+1))^2} =…
Lukas Zamora
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Summation of a series with continuous increase

I am working on a summation of series. Here is the Series 1- 10 0, 10 , 20, 30, 40 , 50, 60 ,70, 80, 90 11-20 100,120,140,160,180,200,220,240,260,280 21-30 300,340,380,420,460,500,540,580,620,660 As one can see, first diff is 10, second is 20 then…
Muhammad Umar
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Is a One-Time Pyment Better Than a Saving Plan (Stocks, Funds)?

My friend and I are theorizing whether or not it is always better to make a one-time payment into a fonds, stock title or whatever instead of a savings plan IF the total interest over a period of time is positive. Example: Here, we have a total…
Alon
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Alternative formula for number of diagonals in a polygon?

While teaching a secondary school student stochastic I found that the sum $$\sum\limits_{n=1}^N(n-2)$$ is equal to the formula used to calculate the number of diagonals in a polygon with N sides: $$\frac{N(N-3)}{2}$$ Can anyone explain why this is…
Luc von Niederhäusern
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How to calculate the number of elements in all subsets of a set

There are $2^N$ subsets of a set. For instance the set $\{ 1, 2 \}$ has the following subsets: $\{ \}$ $\{ 2 \}$ $\{ 1 \}$ $\{ 1, 2 \}$ I'm trying to calculate the total number of elements in all of the subsets. In the above example, it would be…
screeb
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Compute difficult sum

How can I compute this sum? $\sum_{k=0}^\infty\left[\frac{(2k+1)!}{(2k+b)}-\frac{1}{(2k)!(2k+b)}-\frac{(2k)! }{(2k+b+1)}+\frac{1}{(2k+1)!(2k+b+1)}\right]$ I see the $k=0$ term is equal to zero, but that's about as far as I've gotten. Thanks! $b$…
hodop smith
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