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

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2
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1answer
24 views

Is it possible to find the limit of this?

After recently watching a Numberphile video about a square problem I started thinking about what would happen to the sum of all angles if you had n amount of squares. After a bit of testing, I ...
0
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3answers
39 views

convergence of infinite series $\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)(1+x^3)\cdot\cdot\cdot (1+ x^n)}$

I am reviewing for my final exam, and viewed this question: Decide whether the following infinite sum is convergent for all $x >1$: $$\sum_{n=1}^\infty ...
5
votes
2answers
63 views

Prove that $\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$

Prove that $$\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$$ I tried to look at $$ f_n(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^n $$ And maybe taking it's ...
1
vote
3answers
20 views

Sum of segments inside a right triangle.

I am interested for a problem involving the sum of segments inside a right triangle. Consider a right triangle of hypotenuse $\overline{BC}$ and catheti $\overline{AB}$ and $\overline{AC}$. From the ...
0
votes
1answer
48 views

Is there a summation formula for this equation (contains square roots, and functions within the square root)?

I am trying to solve a summation formula that is quite complex. However, to make the "answering" process for you guys easier I'll isolate the part I am having trouble with... The equation is as ...
1
vote
0answers
44 views

Other Patterns in Triples

I have the following 20 triples generated by polynomial distribution: $$\begin{matrix} (2,4,5)&(2,3,4)&(2,3,5)&(1,4,5)&(2,2,4)\\ ...
0
votes
2answers
52 views

Convolution of discrete uniform distributions

For two independent, discrete, uniformly distributed random variables $X$ and $Y$, I wish to obtain the distribution of the sum $Z=X+Y$. I have the densities: $$f_X(x)=\left\{\begin{matrix} ...
1
vote
1answer
81 views

How to prove this Catalan number identity

Catalan number is $\displaystyle C_n= \frac{1}{n+1}\binom{2n}{n}$. How to prove that $$C_{2n-1} = \sum_{k=0}^{n-1}\left(\binom{2n-1}{n-k-1}-\binom{2n-1}{n-k-2}\right)^2$$ for $n\geq 1$. Thank you.
2
votes
1answer
34 views

Estimation of a sum independent of $n$

Suppose $f$ is differentiable on $[0,1]$, $f(0)=f(1)$, $\int_0^1 f(x)dx=0$, $f'(x)\neq 1$. Furthermore, let $g(x)=f(x)-x$, $n\geq 2$ is an integer. Show that ...
2
votes
4answers
96 views

Does $\sum_{n=1}^{\infty}\frac{\cos\left(\frac{n\pi}{2}\right)}{\sqrt{n}}$ converge?

Does the following series converge? $$\sum_{n=1}^{\infty}\frac{\cos\left({\frac{n\pi}{2}}\right)}{\sqrt{n}}$$ The $\cos$ function: alternates between (-1) and 1 for every $n$ that is even. ...
0
votes
1answer
25 views

Product of sums equal to sum of products

Is $(\sum_k x_k)(\sum_i y_i)(\sum_j z_j)(\sum_l a_l) = \sum_{kijl}x_ky_iz_ja_l$ with $\sum_{kijl} = \sum_k\sum_i\sum_j\sum_l$?
4
votes
3answers
121 views

How to prove combinatorial identity $\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose s}{s\choose m-s}$?

The following combinatorial identity have been verified via maple, but I can not prove it. Who can prove it without WZ mehtod? $$\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose ...
1
vote
2answers
102 views

How prove this identity$\sum_{k=0}^{n}\binom{2k}{k}\binom{n+k}{2k}(s-t)^{n-k}t^k=\sum_{k=0}^{n}\binom{n}{k}^2s^{n-k}t^k$

Today I see a paper,and this author say it is easy to have this identity.But I take sometimes to prove it,and I can't prove it. show this following identity holds for any real $s$ and $t$ and any ...
1
vote
3answers
73 views

Prove that $1+2^1+2^2+\ldots +2^n=2^{n+1}-1$ using induction

For all integers $n\ge 1$ prove the following statement using mathematical induction. $$1+2^1+2^2+\ldots +2^n=2^{n+1}-1$$ The first part of the question ask me to prove the base step: So I set ...
1
vote
4answers
76 views

How to sum $\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2}$?

Does anyone know the general strategy for summing a series of the form: $$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2},$$ where $a$ is a positive integer? Any hints or ideas would be great!
0
votes
2answers
24 views

Generating functions and central binomial coefficient

How would you prove that the generating function of $\binom{2n}{n}$ is $\frac{1}{\sqrt{1-4y}}$? More precisely, prove that( for $|x|<\frac{1}{4}$ ): ...
0
votes
1answer
26 views

Determine the convergence of the following series

$$\sum_{k=0}^\infty {3^{k\ln k} \over {k^k}}$$ I need to determine the convergence of this series. I know it diverges, but how do I prove this?
0
votes
0answers
38 views

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: $$\sum_{i=1}^n\frac{a}{b-x_i}$$
1
vote
1answer
32 views

Summation with two running indices

I don't understand the notation of the following summation. $$ \sum_{i,j=1}^m \gamma_i \cdot \beta_{ij} \cdot \alpha_j$$ I first thought $ i, j $ would be increased simultaneously, but that would ...
1
vote
1answer
43 views

Finite natural summation that leads to double exponential results

We know that $$f(n)=\sum_{i=0}^n\binom{n}{i}=2^n$$ and $$g(n)=\sum_{i=0}^ni\binom{n}{i}=n2^{n-1}.$$ Are there any finite natural sums that lead to $2^{2^n}$ or $2^n2^{2^{n-1}}$ results other than ...
1
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3answers
42 views

What does this summation notation denote?

I am having trouble figuring out how this notation works, specifically how the intersection relates to the rest of the summation. It's just stuck there after it. I would greatly appreciate any help ...
1
vote
1answer
32 views

Series Solution to Differential Equation

Given the series $$1 + \sum_{k = 1}^{\infty} \frac{\beta(\beta - 1) ... (\beta - k + 1)}{k!} x^k$$ how can I find a differential equation for which this series is a solution? I don't have any idea ...
2
votes
1answer
35 views

Need to know why $\sum_{k=0}^{\infty}kr^{k} = \frac{r}{(1-r)^{2}}$

Working on a Stat problem where I must find $E(x)$ of $f(x)=\left(\frac{1}{2}\right)^{x+1}$ for $x=0,1,2,\cdots$ I have, ...
2
votes
1answer
39 views

Convergence of $\sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!}$

I need help determining what following series converges to using the ratio test. $$\sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!}$$ It's the end that really has me confused with what to do with the ...
1
vote
2answers
44 views

Question on the sum $\sum_{n=1}^{\infty}\frac{x^n}{n} = -\ln(1-x)$

$f(x) = \displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n} = x + \frac{x^2}{2} + \frac{x^3}{3} + ... = -\ln(1-x)$ for $|x| < 1$. $f'(x) = \displaystyle\sum_{n=1}^{\infty}x^{n-1} = 1 + x + x^2 + x^3 ...
0
votes
1answer
18 views

Some trouble with algebra using logarithms and summations

I'm having some embarrassing trouble with algebraic manipulation. I have the function $$f(y) = y^Tx-\log\sum_{i=1}^ne^{x_i}$$ and for each $i = 1,2,\ldots,n$ $$y_i = {e^{x_1} \over ...
1
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0answers
31 views

Closed forms for two times series similar to geometric series, but with additional power

Does anyone know a close form solutions to any of the following time series? (approximate upper bounds might as well work). $$ \sum_{k=1}^T \frac{1}{2^{k^2}} $$ or $$ \sum_{k=1}^T k ...
-1
votes
1answer
46 views

Prove the formula for the sum of consecutive cubes [duplicate]

$$\sum_{k=1}^n k^3=\frac{n^2 (n+1)^2}{4}$$ Please help
3
votes
1answer
27 views

proving a limit of a series with a sum [duplicate]

I just can't find a way to prove it.
2
votes
3answers
79 views

Question about $\displaystyle\sum_{n=1}^{\infty}\dfrac{|\sin(n)|}{n}$. [duplicate]

In several places on this site the sum $\displaystyle\sum_{n=1}^{\infty}\dfrac{\sin(n)}{n}$ has been discussed as a generalized alternating series, which therefore converges. I am curious about the ...
4
votes
3answers
46 views

Show $\frac{1}{2}\sum_{i\neq j}S_{ij}=\sum_{i<j}S_{ij}$

I am looking to prove the following $$ \frac{1}{2}\sum_{i\neq j}S_{ij}=\sum_{i<j} S_{ij},\qquad S_{ij}=S_{ji}. $$ I am not sure how to understand why it works. Thanks
1
vote
0answers
50 views

Solve summation

I am wondering how one can solve generic 'summations' (or actually: if the idea I had to use integrals is correct): To bring it to the point I have the following sum equation: $$\sum_{i=0}^{19} ...
1
vote
0answers
38 views

Help me finding closed form of sum of 4 elements

I've been reading Wilf's Gfology and tried to calculate some complicated sum Let's say $0<k \le n$ $$f(k,n) = \sum_{i} i(-1)^i \binom{n}{i} \binom{i}{k-i} $$ I will write down my calculations, i ...
3
votes
1answer
58 views

Simplifying the sum $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$

How can I simplify the expression $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$? $x$ is a vector of numbers of length $n$, and I am trying to prove that the result of the expression above is ...
0
votes
1answer
28 views

How do I prove this by induction? [duplicate]

thank you for taking the time to help me with the question. I am struggling to use proof by induction for this formula: $$\sum_{k=0}^{n}k\times k! = (n + 1)! - 1$$ So far, I came up with: $$S(n) = ...
1
vote
2answers
25 views

How to rewrite double sum in matrix operation?

I have a double sum $\sum_{i=1,j=1}^n \alpha_i \alpha_j y_i y_j(x_i,x_j),\ x_i \in R^{d},\ y_i \in R,\ \alpha_i \in R $ How it can be rewritten in terms of vectors and matrices operations?
3
votes
1answer
79 views

Is there a metric in which $1+2+3+4+\cdot$ converges to $-\frac1{12}$?

It is well known that the sum $1+2+3+4+\ldots$, which tends to infinity in the regular sense, can be assigned the value $-\frac{1}{12}$ by different means, e.g., zeta regularization or different ...
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votes
0answers
16 views

Changing the order of summation [duplicate]

In double sums, what are the conditions necessary for being able to switch the order of summation, in the cases of 2 infinite sums, 1 infinite sum and 1 finite sum, and 2 finite sums.
1
vote
1answer
57 views

Closed-form expression for a sum of reciprocals of factorials [closed]

Is there a closed-form expression for the finite sum $$\sum_{s=1}^{2^{n-1}}\frac1{(s-1)!}$$ as a function of $n$?
3
votes
2answers
109 views

Deciding whether series containing $a_n$ are convergent knowing that $\lim_{n\rightarrow\infty}\frac{\frac{(-1)^n}{\sqrt{n}}}{a_n}=1$

We know the following thing about sequence ${a_n}$: $$\lim_{n\rightarrow\infty}\frac{\frac{(-1)^n}{\sqrt{n}}}{a_n}=1$$ And now the problem asks us whether it's true for every such $a_n$ that: ...
3
votes
0answers
30 views

Finding the length of an elliptical spiral

Okay, i had a very strange thought, it was "Is it possible to find the length of an elliptical spiral whose major and minor axes were decreasing?" Like for example lets say that $$ \frac{a}{b} = n ...
8
votes
1answer
159 views

Show all roots of $\sum_{k=0}^n 2^{k(n-k)} x^k$ are real (December 6, 2014 Putnam problem)

Show that for each positive integer n, all roots of the polynomial $\sum_{k=0}^n 2^{k(n-k)} x^k$ are real numbers. I have no idea where to start. From this year's Putnam, problem B4.
-1
votes
2answers
33 views

Find closed form for $2 \times (n + (n + 1) + \cdots + (2n - 1))$

How can we find a closed form for this sum: $$2 \times (n + (n + 1) + \cdots + (2n - 1))$$? Example: $$(7+6+5+4)+(4+5+6+7)=\frac{3}{4} 8^2 - \frac{1}{2} 8$$
1
vote
2answers
27 views

Long summation question, including sets

I have a really long question I'm absolutely stuck on, I don't even know where to begin: Given: $n \in \mathbb{Z}, \geq 2$ let $S$ be the set of all nonempty subsets of {2,3,...,n}. For each $S_i ...
0
votes
3answers
29 views

What is the relationship between the some of the first n natural numbers and the number of unique pairs in n+1

The sum of the first $n$ natural numbers is $\displaystyle\frac{n}{2}(n+1)$ $\displaystyle {n+1 \choose 2}$ is $\displaystyle\frac{n}{2}(n+1)$ Obviously they are equal, but how should I think ...
1
vote
3answers
81 views

Evaluating $\sum_{n=1}^{\infty}\frac{6}{n(n+1)(n+2)}$

Show that series $$\sum_{n=1}^{\infty}\frac{6}{n(n+1)(n+2)}$$ converges by simplifying its sequence of partial sums and find its sum. I don't have much detail but this all I have: ...
0
votes
1answer
26 views

A question about non consecutive sum writing

I am very confused and I thank you in advance for your help about a very stupid question. I would write a summation sequence of non consecutive primes $a,b,c,d...$ greater than $3$ with a distance of ...
3
votes
2answers
50 views

Average number of trials until drawing $k$ red balls out of a box with $m$ blue and $n$ red balls

A box has $m$ blue balls and $n$ red balls. You are randomly drawing a ball from the box one by one until drawing $k$ red balls ($k < n$)? What would be the average number of trials needed? To ...
7
votes
2answers
171 views

Closed-form of $\sum_{n=0}^\infty\;(-1)^n \frac{\left(2-\sqrt{3}\right)^{2n+1}}{(2n+1)^2\quad}$

The following question is purely my curiosity. During my calculation to answer @Chris'ssis's question in chat room I encountered this series $$\sum_{n=0}^\infty\; ...
0
votes
2answers
32 views

Ratio of 2 Sums of products of binomial coefficients

I want to prove that for $k \ even, 0 \leq k<n, n\in \mathbb{N}$: $-\frac{1}{(2n-3-k)(k+2)}\sum \limits_{i=0}^{k} \frac{(-1)^{i} 2^{i} (2n-2-i)!}{(n-1-i)!i!(k-i)!}=\sum \limits_{i=0}^{k+2} ...