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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Trick to solving this Summation?

I am currently following some slides I found to try and learn how to find the average case complexity for some algorithms. I got stuck when having to handle this summation though: $$\sum_{i=1}^k i*2^i $$ According to the slides: it can be shown…
xdv
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Alternating sum of reciprocals

Find the infinite sum: $\frac{1}{(1)(2)}+\frac{1}{(3)(4)}+\frac{1}{(5)(6)}+\cdots$. I thought of expanding it as $\frac{1}{1}-\frac{1}{2}+\frac{1}{3}- \cdots$, but it would yield nothing useful as this runs into the non-convergence issue.
The Math Troll
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The summation of a 'sinc' like series

I would like to know how to solve or approximately express the summation below $$\sum_{n=0}^{N-1}\frac{e^{i2\pi\alpha n}-e^{i2\pi(\beta n-\gamma)}}{i2\pi((\alpha-\beta)n+\gamma)} $$ where $\alpha$, $\beta$, and $\gamma$ are all reals, $i =…
Guanghong
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Simplifying $\sum\limits_{k=0}^{\infty}\frac 1{2^{k+1}}\sum\limits_{n=0}^{\infty}\binom kn\frac 1{2(n+1)(3n+1)}$

Question: Is there a way to simplify $$\sum\limits_{k=0}^{\infty}\dfrac 1{2^{k+1}}\sum\limits_{n=0}^{\infty}\binom kn\dfrac 1{2(n+1)(3n+1)}\tag{1}$$ Into a single summation symbol? $\sum\limits_{k=0}^{\infty}\text{something}$ I inputed it into…
Crescendo
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Proof that $\sum_{x=0}^{y} ({{y!} \over {x! (y-x)!} })=2^y$

My teacher substitutes for this $\sum_{x=0}^{y} ({{y!} \over {x! (y-x)!} })$ by $2^y$, so I tried to use the (Mathematical induction) to prove it (My teacher did not ask me to do that, however I want to do it only to make sure for this statement) My…
Dima
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Determine: $S = \frac{2^2}{2}{n \choose 1} + \frac{2^3}{3}{n \choose 2} + \frac{2^4}{4}{n \choose 3} + \cdots + \frac{2^{n+1}}{n+1}{n \choose n}$

We are given two hints: consider $(n+1)S$; and use the Binomial Theorem. But we are not to use calculus. My consideration of $(n+1)S$ goes like this: \begin{align*} \sum\limits_{k=1}^{n}\frac{2^{k+1}}{k+1}{n \choose k} &=…
Adam
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Expressing $1-\sum_{k=1}^{r}(-1)^{k-1}\binom{r}{k}$ as one sum?

Hi how can i make this expression into one sum(ie put the 1 inside) $1-\sum_{k=1}^{r}(-1)^{k-1}\binom{r}{k}$ I know that $\sum_{k=0}^{r}(-1)^{k}\binom{r}{k}$ comes out but i cant follow the steps ?
asddf
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How to calculate this sum.

I am trying to solve an excercise and i come across the following sum $ \sum_{k=1}^n \frac{(k+1)(k^3-2k+2)}{k(k+2)} $ I put it in Wolfram alpha and it says that it is equal to: $\frac{n(2n^4+6n^3+2n^2+3n+11)}{6(n+1)(n+2)}$ but how can i prove this?
Andreas Ch.
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Finding out the summation of maximum element of subsets of a set

Let $S={1,2,...,j}$ be a set.For every nonempty subset of $A$ of $S$,let $m(A)$ denote the maximum element of $A$.Then find $$\sum_{\text{over all subsets of S}} m(A)$$ By seeing the answer I could easily prove inductively that the summation is…
Navin
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Prove that $ \sum_{m=0}^{{\rm min}\{n,l\}} \binom{l}{m} \frac{n!}{(n-m)!} x^{l-m} = \sum_{m=0}^{l} \binom{l}{m} (n-l+m+1)_{l-m} x^{m}$

How to show that for any non-negative integers $n$ and $l$ two sums below are equal: $$ \sum_{m=0}^{{\rm min}\{n,l\}} \binom{l}{m} \frac{n!}{(n-m)!} x^{l-m} = \sum_{m=0}^{l} \binom{l}{m} (n-l+m+1)_{l-m} x^{m}, $$ where $(a)_s= a(a+1)(a+2)⋯(a+n−1)$…
z.v.
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Deriving summation formulas

I'm working through Spivak's Calculus (Fourth Edition). In chapter 2, problem 6: The formula for $1^2 + ... + n^2$ may be derived as follows. We begin with the formula $(k+1)^3 - k^3 = 3k^2 +3k +1$ Writing this formula for $k = 1, ... , n$ and…
Ahmed99
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How to evaluate the sum $\sum_{i=1}^{2n}\sum_{j=i}^{n+i}\sum_{k=1}^{j} n $?

I am trying to figure out the time complexity of the a given Algorithm, but using summations. Can anyone help get started on the inner summation, because I haven't come across such summation yet, and we just began solving these. …
Kono Dio Da
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Reverse the bounds of a summation

I'm trying to figure out how reversing the sum works. I know it involves changing or switching variables, but I'm stuck on how to do that. How do I go from $c + \sum\limits_{i=0}^{n-2} d(n - i)$ to $c + \sum\limits_{i=2}^{n} d(i)$?
Jasmine
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How would you write this in Sigma notation

I stumbled upon this expression:$$p_nq_mx^{m+n}+(p_{n-1}q_m+p_nq_{m-1})x^{m+n-1}+(p_{n-2}q_m+p_{n-1}q_{m-1}+p_nq_{m-2})x^{m+n-2}+\ldots+p_0q_0\tag1$$ And I'm wondering if there is an easier way to represent $(1)$ using the sum notation Sigma:…
Crescendo
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Working on this summation proof

$$\sum_{i=1}^n t^{i-1}$$ I am stuck with the proof of this equality.
HarveyMudd
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