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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Summation to Equation

I have a summation and I want to be able to find the sum for given $n$ without having to go through $1,\dots,n$. $$\sum_{x=1}^{n - 1}x+300\cdot2^{x/7}$$ It's been awhile since I've done summations and I can't figure this one out. Also, this isn't…
Stripies
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How do we justify splitting a sum over $1 \le j < k+j \le n$ into two sums $1 \le k \le n$ and $1 \le j \le n-k$?

Can anyone justify the following summation manipulation? $$\sum_{1 \le j < k+j \le n} \frac{1}{k} = \sum_{1 \le k \le n}\,\sum_{1 \le j \le n-k} \frac{1}{k}$$
user156100
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Summation of series

If the sum $$\sum_{n=0}^{2011} \frac{n+2}{n!+(n+1)!+(n+2)!}$$ can be written as $$\frac{1}{2} - \frac{1}{a!}$$find the last three digits of a. I have reduced the given expression to $$\frac{1}{(n+2)(n)!}$$ and I think I will have to use the method…
user140161
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A sum involving binomial coefficients and a simple fraction

Let $a_1$ and $a_2$ be real numbers. Let $n_1$ and $n_2$ be positive integers. Finally let $\theta$ be a real number which is different from a negative integer. By the generalizing the result from Another sum involving binomial coefficients. and…
Przemo
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How to simplify this summation: $\frac{2\sum_{k=0}^n2^n}{n+1}=2^{n+1}$?

So I saw an earlier post where they had this equation here. $\frac{2\sum_{k=0}^n2^n}{n+1}=2^{n+1}$? However, I do not know how they did this? Am I missing something?
Chinedum Ukejianya
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What is the sum of a finite series when the variable is in the denominator?

I would like to know if there is a closed form way of writing this sum please: $$\sum_{i=1}^n\frac{a}{b-x_i}$$
Esmerelda
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Is there any way to approximate a sum of square roots

I am trying to calculate a sum of square roots $\sum\limits_{i=1}^n \sqrt{a + i}$ and after some struggling and googling I gave up on this. Is there any way to get a closed formula for this sum (actually even approximation with epsilon $10^{-4}$…
Salvador Dali
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Exact closed form expression of $(2^0+...+2^n)+(2^1+...+2^{n+1})+...+(2^n+...+2^{2n})$

Exact closed form of this expression $(2^0+...+2^n)+(2^1+...+2^{n+1})+...+(2^n+...+2^{2n})$ I assume this means there is just one $2^0$ and one $2^{2n}$ and a double of all the terms in between?
Benjamin
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Why is this nested sum formula true

I've been trying to get this sum: $\sum_{i}^{n} \sum_{j=0}^{n-i}j$ into a closed formula but couldn't really understand how to "unpack" that nested sum. It occured to me that the answer is: $$\sum_{i=1}^n \left(\sum_{j=0}^{n-i} j\right) = \frac16 n…
wannabe programmer
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Convergence of a summation

How do I find out what this summation converges to? I dont even know how I'd start o_o I assume the first part converges to infinity but dont know how the cos works in this case. $$\sum^{\infty}_{k=1} 3^{-k}\cos{k}$$
Tomas Mas
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sum of two equal digit numbers vs. sum of those digits

if I take 5688+6984=12672 then sum the result 1+2+6+7+2=18 then sum that result 1+8=9. vs. this. same digits from above. 5+6+8+8+6+9+8+4=54 then sum that result 5+4=9. using this method where the original two are always the same length, four digits…
shawn
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find the sum of series

I have problem with finding sum of series: 1)$\displaystyle\sum_{n=k}^{2k-1}\frac{n}{2^n}=?$ 2) $\displaystyle\sum_{n=0}^{k-1}n(\frac{4}{3})^n=?$ I have some idea to 1) ot write it as…
Mario
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a complicated Sum

I don't know how i can calculate this complicated multivariate sum : $$ S(l,k)=\sum_{|m|=k} s_{l,m}=\sum_{|m|=k} l(l-1)(l-2)\dots(l-m+1) $$ Where $m=(m_1,\dots,m_n)$, $l=(l_1,\dots,l_n)$, and $k$ a non-negative integer and…
Hamza
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Rewriting an infinite sum

Rewrite the given expression as a sum whose generic term involves x^n: $$ x\cdot\sum_{n=1}^{\infty}(n a_n x^{n-1}) + \sum_{k=0}^{\infty}(a_k x^{k} ) $$ I get a sum starting at one: $$ \sum_{n=1}^{\infty}(n a_n x^{n}) + \sum_{n=1}^{\infty}(a_n x^{n}…
V.Vocor
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Help explaining sum notation

So, I have the question and I also have the answer. Need to prove: And here is the answer Can you please explain the steps. They are in the second picture but I do not understand where they are coming from. Thank you in advance.
helloima
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