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 sequences-and-series for sums of infinite series and questions of convergence; use summation for questions about finite sums and simplification of expressions involving sums.

17770 questions

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Given its definition, the formula seems incorrect (Normalized Distance-based Performance Measure)

Background information My question is mostly mathematic but requires a bit of knowledge, which I will introduce here. I am currently working through a microsoft paper (p. 20) about the evaluation of recommender systems. There is one specific block…

summation

asked Feb 12 '14 at 11:22

Xiphias

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1 answer

Proving Summations

I'm unsure of how to continue in my proof. How can I prove the follow through induction: $\sum\limits_{k=66}^n {k-1 \choose 65} = {n \choose 66}$ where $n \geq k \geq 66$ Basis:Let $n=66$. $$\sum\limits_{k=66}^{66} {66-1 \choose 65} = {66 \choose…

summation

asked Jan 26 '14 at 23:11

GivenPie

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1 answer

How do I calculate a sum containing binomial coefficients?

Let $a \in (0,1)$, $l\le k-3-p$, $p\ge0$. The question is to show the following identity. \begin{equation} \sum\limits_{l_1=l+2}^{k-1-p} C^{k-l_1-1}_{p} \frac{l_1^p}{a^{l_1}} = \frac{\left(\sum\limits_{s=p+1}^{2p+1} a^{s-k} \mathcal{A}_s\right) + …

summation

asked Jan 08 '14 at 19:17

Przemo

11,331

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3 answers

How to simplify the following equation?

I have an equation for $\sum\limits_{i=1}^ni2^i=\text{res}$ the equation is converted to $$ (n-1)2^{n+1}+2=\text{res} $$ I know the value of res but how to get value of $n$ . It's OK to have an approximate solution.

summation

asked Dec 31 '13 at 02:24

Madan Ram

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4 answers

How to calculate the sumation of a function in one step?

I'm building a Hierarchical Agglomerative Clustering algorithm and I'm trying to estimate the time the computer will take to build a hierarchy of clusters for a given set of samples. For $m$ samples, I have to calculate $m-1$ levels in a binary…

summation

asked Dec 23 '13 at 08:42

Daniel Cerecedo

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2 answers

Closed form of summation with n+1 on top?

How do you find the closed form of summations such as: $$\sum\limits_{k=1}^{n+1} k$$

summation

asked Dec 09 '13 at 00:19

camelbak

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1 answer

Why is $\sum_{i=0}^{n-5} 4(n-i-5)^3 = 4 \sum_{i=5}^n (n-i)^3$?

Why is this equivalence true? $$\sum_{i=0}^{n-5} 4(n-i-5)^3 = 4 \sum_{i=5}^n (n-i)^3$$ With the first sum I would make an index shift by 5 and would get: $$4 \sum_{i=5}^n (n-i-10)^3.$$ My questions are thus: Why is the first equivalence true? What…

summation

asked Dec 08 '13 at 18:33

Elternhaus

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4 answers

Simplifying sum with factorial in denominator

I am trying to to find the sum of this series: $\sum_{x=0}^{\infty}\frac{x^2(1/2)^xe^{-1/2}}{x!}$, but I am stuck because I don't know how to deal with the factorial in the denominator. Is this perhaps somehow related to a Taylor series expansion?

summation

asked Dec 07 '13 at 19:20

Luchia

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1 answer

Sums of Geometric Progressions

Let Find an expression for $S_2$ in terms of $S_0$ and $S_1$. Do not need to simplify. If someone could explain this, that would be a real life saver.

summation

asked Dec 05 '13 at 06:04

Neal

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2 answers

Exponential sum simplification

I've got this: $$\sum_{k=1}^{n}\frac{1}{2^{2k}}$$Wolfram Alpha already simplified it for me as $\dfrac{2^{2n}-1}{3\cdot2^{2n}}$, but he hasn't told me why.

summation

asked Dec 02 '13 at 19:56

k5f

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1 answer

First Sum of Zetas

Can I simplify the sum: $\zeta(1) +\zeta(2) +...+\zeta(n)$ . Whereas $\zeta$ denotes the zeta function. I want to find the limit involving this sum but need it simplified. Thanks.

summation

asked Nov 27 '13 at 17:14

DeepSea

77,651

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3 answers

Finding a minimum error in sum

I have the sum $$S=\sum_{n=1}^{\infty}(-1)^n(\frac{n}{n^2+1})$$ And Im trying to find the sum $S_n$ where the error approxomating $S$ is less than $\frac{1}{1000}$. I calculated it at found what I tought was the right answer, but it turned out to be…

summation

asked Nov 23 '13 at 05:19

user100620

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1 answer

Differentiating a sum to infinity

Let $G:[0,1] → R$ be given by: $G(s)= \sum s^n$ (this sum is from $0$ to $\infty$) I have to evaluate $G'(s)$ in two different ways: Differentiate the terms in the sum terms by term. Sum the series on the right hand side of equation and then…

summation

asked Oct 30 '13 at 15:47

Rich A

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4 answers

How to guess that $ \sum_{i=1}^{n}3^i = \frac{3}{2}(3^n - 1)$

As in title how do you guess that $ \sum_{i=1}^{n}3^i = \frac{3}{2}(3^n - 1)$? I have homework about solving recurrence relations and using iterate method I can find that…

summation

asked Oct 19 '13 at 12:28

Machiaweliczny

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1 answer

A closed form for $\sum_{k = 1}^{\infty} k^{-k}$?

Is there any closed form for this expression? $$\displaystyle\sum_{k = 1}^{\infty} k^{-k}$$ I got this while playing with my scientific calculator, so I really have no idea about how I could find one.

summation

asked Oct 08 '13 at 13:52

Soham Chowdhury

1,797

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