Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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0
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0answers
2 views

Does a piece wise function inside a sum make sense (fractional bias)?

I'm looking for a good way to express the calculation for the fractional bias. It is used to compare simulated values ($s$) with observed values ($o$) and pretty simple to calculate: build the ...
0
votes
1answer
13 views

How to simplify an expression where the summation is over all subsets of a given set?

I want to simplify this expression: $\sum\limits_{\emptyset \ne I \subseteq \mathbb{N}_k}(-1)^{|I|-1}|A_I|+|A_{k+1}|-\sum\limits_{\emptyset \ne I \subseteq \mathbb{N}_k}(-1)^{|I|-1}|A_I \cap ...
12
votes
4answers
239 views

Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Show$$\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$$diverges. Just going down the list, the following tests don't work (or I failed at using them correctly) because: $\lim ...
15
votes
3answers
358 views

How prove this sum $\sum_{n=1}^{\infty}\binom{2n}{n}\frac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}$

show that $$\sum_{n=1}^{\infty}\binom{2n}{n}\dfrac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}=5+4\sqrt{2}\left(\log{\dfrac{2\sqrt{2}}{1+\sqrt{2}}}-1\right)$$ where ...
3
votes
1answer
24 views

Measuring sums of complex alternating series

Suppose we have functions $$f(x) = \sqrt{x}, \space g(f) = \frac{df}{dx}+\frac{d^2f}{dx^2}+\frac{d^3f}{dx^3}\space ...$$ Applying function f(x) to g(f) we get: $$g(f(x))=\frac{1}{2}x^{-\frac{1}{2}} - ...
0
votes
1answer
37 views

How to calculate the sum of combinatorial numbers

For my work on an almost completely unrelated field I came across the following formula. I know that I have learned that all in high school. But since this is more than 15 years ago in which I never ...
2
votes
2answers
39 views

is there a generating function for $H_{2n}$?

I have been wondering if anyone knows if there is a generating function for harmonic series of the form $H_{2n}$?. That is, we are familiar with ...
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0answers
203 views
+50

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
0
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1answer
24 views

Easy question regarding this proof

I do not understand a small step in a proof I'm reading at the moment. Why are the following things equal? $$\sum_{k=1}^{n} \frac{1}{2k-1} - \frac{1}{2} \sum_{k=1}^{n} \frac{1}{k} = \sum_{k=1}^{2n} ...
2
votes
1answer
58 views

simplifying a triple sum of products of binomial coefficients

Right now I have a horribly-looking triple sum ($x,y,z$ are non-negative integers and $x+y+z=N$): $$ W_{12}(x,y)=\frac{x}{N}\sum_{l=0}^{x-1}\sum_{l'=0}^{y}\sum_{l''=0}^{z}{x-1 \choose ...
0
votes
2answers
43 views

double summation problem $\sum^5_{i=1}i \times \sum^5_{j=1}j =…$ please check

(I) $\sum^5_{i=1}i \times \sum^5_{j=1}j = 1 \times (1) +1 \times (2) + \cdots +1\times (5) +2\times (1)+2\times (2) +\cdots + 2\times (5) + 3\times (1) + 3\times (2) + \cdots +3\times (5) + 4\times ...
4
votes
2answers
35 views

$\sum_j e^{i\phi_j}$ vs $\sum_j e^{ip\phi_j}$

Let $\phi_j$ be a collection of angles. If $p$ is a positive integer, how is the sum $\sum_je^{i\phi_j}$ related to $\sum_je^{ip\phi_j}$?
3
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0answers
68 views

Closed form $\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}}$ and $\sum_{n=2}^{\infty} \frac{n}{\ln^n{n}}$

Apologies if this has been asked before, but I was playing around with Wolfram Alpha and got approximations but not closed forms for $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
1
vote
5answers
127 views

$\sum_{k=0}^{n}\frac{1}{(k+1)(k+2)}\binom{n}{k}=?$

I was asked to find a closed formula for the sum $$\sum_{k=0}^{n}\frac{1}{(k+1)(k+2)}\binom{n}{k}$$ could anyone give me an advice on how to get started?
7
votes
4answers
378 views

Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$

The other day a friend of mine showed me this sum: $\sum_{k=0}^n\binom{3n}{3k}$. To find the explicit formula I plugged it into mathematica and got $\frac{8^n+2(-1)^n}{3}$. I am curious as to how one ...
1
vote
3answers
172 views

Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$

I need hint proving this $$\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$$
1
vote
2answers
92 views

How to find $\sum_{r=1}^{n} r^2\cos {(r\theta)}$

How do you find the sum $$S(\theta)=\displaystyle\sum_{r=1}^{n} r^2\cos {(r\theta)}$$? I observed that if $f(\theta)$ is $\sum\cos {(r\theta)}$, then $S(\theta)=-f''(\theta)$. Help will be ...
5
votes
1answer
136 views

Prove $(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j} $

I stumbled upon the identity $$(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j}. $$ The right-hand side is a polynomial. ...
0
votes
2answers
21 views

How to get a partial sum formula

Let S denote sum from 1 to n of (k-1)/k! . I tried obtaining a partial sum formula, but I couldn't get too far. WolframAlpha comes with quite a simple form, but I fail to see how they got there . Can ...
4
votes
2answers
85 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
3
votes
1answer
43 views

investigate $\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$

I need to investigate the series (Hence, when the series converges and when the series converges absolutely depending on $\alpha$). $$\sum\limits_{n\ge2}{\frac{(-1)^n}{n^\alpha \ln n}}$$ For ...
1
vote
1answer
30 views

evaluating a sum using Cauchy condensation test

Let $$\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$$ I want to check if the sum is converges absolutely. Hence, we need to check the convergence of $$\sum\limits_{n\ge1}{\frac{1}{n^\alpha \ln ...
6
votes
1answer
183 views

find the sum $\sum_{k=0}^{n} k\binom{n}{k}^2p^k$

I need to find the sum $$\sum_{k=0}^{n} \binom{n}{k}^2kp^k,$$ for an integer $n$ and $0<p<1$. Mathematica would only return $$\sum_{k=0}^{n} k p^k \binom{n}{k}^2 = n^2 p \ _2F_1(1-n, 1-n, 2, ...
6
votes
3answers
110 views

How to find sums like $\sum_{k=0}^{39} \binom{200}{5k}$

How do I find sums like these?-- $$S=\displaystyle\sum_{k=0}^{39} \dbinom{200}{5k}$$ that is, when there is a summation of binomial coefficients, but with jumps of some terms..?
0
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2answers
76 views

The sum of $1^2+7^2+13^2+\cdots+n^2,$ where $ n =1 \mod6 $

Find the sum of this progression in the terms of $n$ $$1^2+7^2+13^2+\cdots+n^2,$$ where $ n =1 \mod6 $. Are Bernoulli's numbers involved in this? Please help.
2
votes
1answer
220 views

Sums of Primitive Roots and Quadratic Residues when $p \equiv 3\pmod 4$

Define $$R_{p}=\{ r \mid r: \text{primitive root of p}, 1 \le r \le p \}$$ and also $$Q_{p}=\{ a \mid a: \text{quadratic residue of p}, 1 \le a \le p \}$$ $$Q_p^c=\{a \mid a: ...
1
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1answer
59 views

$\sum\limits_{i,j\in[|1,n|]}\frac{\max\left(i,j\right)}{\min\left(i,j\right)}$

I'm trying to do this sum $\sum\limits_{i,j\in[|1,n|]}\frac{\max\left(i,j\right)}{\min\left(i,j\right)}$ What I have tried : $\sum\limits_{i,j\in[|1,n|]}\frac{\max ...
5
votes
2answers
157 views

What is the closed form for $S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$?

How do we find the following sum (closed form)? $$S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$$ Edit What about $$S=2\left(\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ...
7
votes
3answers
253 views

How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}} $

Can we find $$ \sum_{k=0}^{n} \sqrt{\binom{n}{k}} \quad$$ This problem asked me my friend about a year ago, but I didn't know how to attack problem. Now, I am interesting in solution. Any suggestion? ...
3
votes
2answers
81 views

Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
0
votes
1answer
25 views

Help with finding the arithmetic mean of all the radii from the center to the edge of an ellipse?

So far I approached this problem computationally, I decided to take all the radii add them up, by distance formula, then divide by the number of radaii. To make the distribution even, I rotated the ...
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0answers
40 views
0
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2answers
56 views

How to find the value of this summation?

How to solve this summation? $$S=\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n4^{4n+1}}$$ I tried it to convert it into a definite integral but wasn't successful. Help will be appreciated.
1
vote
2answers
70 views

Partial sum of binomial

I 'm trying to figure out a closed form solution for the following summation: $\sum_{j=0}^{\omega} j{n \choose j}p^{j}(1-p)^{n-j}$ where $\omega < n$ Is there any closed form solution?
3
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3answers
72 views

Closed form for the sum: $\sum_{n=1}^{\infty}\frac{1}{n(n + 1/3)}$ [duplicate]

I tried using partial fractions to compute the sum of the series $$ \sum_{n=1}^{\infty}\frac{1}{n(n + 1/3)} $$ Another technique is to turn this series into a definite integral of 0 to 1. but do not ...
0
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0answers
41 views

An equation on Catalan number [closed]

Catalan numbers have the form $C_n=\frac{1}{n+1}\binom{2n}{n}$ prove: $C_{n+1}=\sum_{m+k=n}C_mC_k$ I tried to expand $C_n$ but soon get confused..
11
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1answer
7k views

Combinatorial proof of summation of $\sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$

Can you guys help me prove this? There is a way of proving this logically but I was hoping to find a more "mathematical" proof, if possible. $$\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n ...
0
votes
3answers
62 views

Can you illustrate the use of coordinate-free notations that serve as an alternative to Einstein summation notation with an example?

"Abstract index" and "coordinate free notations" are often submitted as alternatives to Einstein Summation notation. Could you illustrate their use using an example? Here's a sum written in ...
1
vote
2answers
30 views

$\sum_{k=1}^m n-k$ and $\sum_{k=m+1}^{2m} k$

i really can't find out why $$\sum_{k=1}^m n-k = -\frac12\left(m^2-2nm +m\right)$$ and why $$\sum_{k=m+1}^{2m} k = \frac12m\, (3m +1)$$ For the first one i really don't know where to start, but for ...
2
votes
1answer
38 views

Evaluating $\sum \sin(ak) x^k / k! $

Mathematica tells me that $$ \sum_{k=0}^\infty \sin(ak) \frac{x^k}{k!} = e^{x \cos (a)} \sin (x \sin (a)). $$ How would I go about evaluating such a series by hand? My first thought is to expand ...
0
votes
1answer
53 views

Hypergeometric function representation

Is it possible to express the following sum in terms of the hypergeometric function $_2F_1$: $$ f(x) = \sum_{n=0}^\infty\frac{(-ax)^n}{n!~\Gamma(b-n)} $$ with $a$ and $b$ constant values ($x>0$ ...
3
votes
1answer
29 views

Inequality containing finite sum.

For what value of k the following inequality holds? $\sum_{i=1}^{n}a_{i}^3<k|\sqrt{\sum_{i=1}^{n}a_{i}}|$ I don't have any idea to solve this.
4
votes
1answer
154 views

Interesting Sum Congruence

Let $5\mid a$, $\gcd(a,b)=1$, and $b\equiv 2\bmod 5$. How can one show that $\sum_{k=1}^{a}k\lfloor\frac{kb}{a}\rfloor\equiv 2\bmod 5$? Similarly, can we show that if instead $b\equiv 3\bmod 5$, then ...
8
votes
3answers
268 views

Proving that $ \displaystyle \gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $.

In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant $ \gamma $. The formula is $$ \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ ...
5
votes
2answers
247 views

Reference for a tangent squared sum identity

Can anyone help me find a formal reference for the following identity about the summation of squared tangent function: $$ \sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+. $$ I ...
2
votes
1answer
77 views

What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$ \frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0 $$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
7
votes
2answers
154 views

Prove that $\displaystyle{\sum_{n=1}^{\infty}}(-1)^{n-1} \dfrac{H_n}{n} = \dfrac{\pi^2}{12} - \dfrac{1}{2}\ln^2 2$

We know that $H_n = \sum_{j=1}^{n}{1 \over j}$. Article in The Sum of Certain Series Related To Harmonic Numbers of Omran Kolba, we have proof of this identity which involves some advanced concepts. ...
1
vote
0answers
34 views

Sum involving binomial coefficients

Exist a closed form for $$\left(-1\right)^{N}\underset{i=1}{\overset{N}{\sum}}\left(-1\right)^{i}\dbinom{N}{i}\dbinom{N+i}{i-1}\,\frac{1}{2i+1}?$$ I think I've to use in some way the formula of the ...
4
votes
3answers
476 views

Does a closed form exist for this summation?

How do I calculate $$\sum_{k=1}^\infty \frac{k\sin(kx)}{1+k^2}$$ for $0<x<2\pi$? Wolfram alpha can't calculate it, but the sum surely converges.
4
votes
0answers
124 views

Proving an equation involving binomial coefficients

Prove that $$\sum_{q=0}^v \binom{v}{q}\frac{q!}{v^{q+1}} = \sum_{q=0}^{v-1} \binom{v-1}{q} \frac{(q+2)!}{v^{q+2}}$$ Thanks. Below are what I have tried: Approach 1: $$\sum_{q=0}^{v-1} ...