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

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0answers
17 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
21 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 ...
0
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1answer
44 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 ...
3
votes
1answer
46 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}} - ...
7
votes
6answers
166 views

Finding $\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $

Help me to simplify:$$\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $$ I got a hunch that it will depend on whether $n$ is a multiple of $6$ and equals to $\frac{2^n+2}{3}$ when $n$ is a ...
2
votes
2answers
54 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 ...
0
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1answer
26 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} ...
4
votes
2answers
38 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 $\sum_je^{ip\phi_j}$?
14
votes
3answers
435 views

Equality of sums with fractional parts of the form $\sum_{k=1}^{n}k\{\frac{mk}{n}\}$

I recently encountered the following equality ($\{\}$ denotes fractional part): $$\sum_{k=1}^{65}k\left\{\frac{8k}{65}\right\}=\sum_{k=1}^{65}k\left\{\frac{18k}{65}\right\}$$ and found it very ...
6
votes
1answer
243 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
2answers
99 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 ...
0
votes
2answers
23 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 ...
3
votes
1answer
50 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
31 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 ...
0
votes
2answers
82 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.
0
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1answer
34 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 ...
3
votes
2answers
88 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 ...
6
votes
2answers
194 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)? $$S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$$
2
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0answers
49 views

Sum and binomials

I have this sum ...
0
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2answers
60 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.
6
votes
2answers
172 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. ...
1
vote
2answers
79 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
votes
3answers
76 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
votes
3answers
70 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
33 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
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1answer
40 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 ...
9
votes
3answers
161 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..?
3
votes
1answer
30 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.
0
votes
1answer
57 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 ($x>0$ ...
1
vote
0answers
40 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
488 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
144 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} ...
2
votes
1answer
47 views

Identity with binomials [duplicate]

Does there exist a closed formula for $$\underset{n=1}{\overset{N-1}{\sum}}\dbinom{N+n}{n}?$$ I've searching on wikipedia but I haven't found this kind of sum.
1
vote
0answers
26 views

Abel's Theorem, alternate proof

I'm trying to solve: Suppose $\sum_{n=1}^\infty a_n$ converges. Prove that: $$ \lim_{r\to1^-}\sum_{n=1}^\infty r^n a_n = \sum_{n=1}^\infty a_n. $$ Hint: Sum by parts. In class, I have seen a ...
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0answers
43 views

Find $\lim a_k$ when $\sum_{k=1}^{n} \frac{a_k}{(n+1-k)!} = 1$, $\forall n \in \mathbb{N}$ [duplicate]

$$ \sum_{k=1}^{n} \frac{a_k}{(n+1-k)!} = 1 $$ A sequence $(a_k)$ satisfies the above expression, $\forall n \in \mathbb{N}$. ($a_1 = 1, a_2 = \frac{1}{2}, a_3 = \frac{7}{12}, \cdots$) What is the ...
2
votes
3answers
42 views

Series question with logarithms

I want to know how to check the divergence of following sum: $\sum_{k=0}^\infty \frac{1}{\sqrt[n]{\log n}}$ I tried to use this result: $ \lim_{n \rightarrow \infty} \frac{1}{\sqrt[n]{\log n}}=1 ...
3
votes
1answer
47 views

limit of a sum of powers of integers [duplicate]

I ran across the following problem in my Advanced Calculus class: For a fixed positive number $\beta$, find $$\lim_{n\to \infty} \left[\frac {1^\beta + 2^\beta + \cdots + n^\beta} {n^{\beta + ...
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0answers
59 views

Showing that two sums are equivalent

given \begin{gather} U_d(x,y,q\mid i_1,\ldots,i_k)=\sum\limits_{n,m\geq0}x^ny^m\sum\limits_{\sigma = i_1\ldots i_k\sigma_{k+1}\ldots\sigma_m\in C_{[d]}(n,m)}q^{v(\sigma)}. \end{gather} show ...
7
votes
2answers
164 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. ...
4
votes
2answers
162 views

Combination of quadratic and arithmetic series

Problem: Calculate $\dfrac{1^2+2^2+3^2+4^2+\cdots+23333330^2}{1+2+3+4+\cdots+23333330}$. Attempt: I know the denominator is arithmetic series and equals ...
0
votes
0answers
40 views

The sum of palindromes from 100 to 900

I'm working with palindromes from $100-999$. I'm having trouble with the step highlighted in red. Can someone explain the algebra to me? Taken from: Discrete and Combinatorial Mathematics: An ...
1
vote
1answer
83 views

If $a + b + c = 0$ prove that

If $a + b + c = 0$, prove that 1)$$ \sum_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 3 $$ 2)$$ \prod_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 1 $$ There is a solution that uses two cubic ...
4
votes
1answer
156 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 ...
2
votes
1answer
33 views

What is Vandermonde's formula with multisets?

I need Vandermonde's formula in multi-set form. I modified the original formula but I get a mess with too many letters everywhere, is there a nice representation? Here's the original: $$ ...
6
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4answers
771 views

What's the purpose of this formula?

Just found this image on the web: Can anyone explain what's the meaning (if any) of this formula? (I did a Google image search but found no answer)
2
votes
0answers
113 views

Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$

For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate $n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$ So the product goes up to $k$ and I ...
1
vote
1answer
59 views

How do we solve complicated summations

I am reading Introduction with Algorithms book and my first doubt arises in time analysis of my first insertions sort's algorithm uses these sigma problems. I learned about this in junior classes but ...
4
votes
6answers
105 views

If $\lim\limits_{x \to \infty} f(x) = 1$, can we have function $f(x)$, such that $\int_0^{\infty}f(x)dx$ converges

I know the Initiative answer, can anyone give a neat answer based on solid reasoning EDIT : $f(x)$ is continuous
0
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4answers
138 views
-3
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2answers
39 views

Calculating two Summation [closed]

Could you help me to calcule this summation? Thanks in advance!