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

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0answers
19 views

Evaluation sum indexed by non decreasing sequences

During solving a problem from probability theory, I've met the following sum to evaluate: $$p_n(N) = \frac{1}{N!}\sum_{0\leqslant k_1\leqslant\ldots\leqslant k_n\leqslant N}\frac{k_1\cdot\ldots\cdot ...
1
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0answers
29 views

Infinite Sum involving Laguerre Polynomials

I would like to simplify (if possible) $$ \sum_{k=0}^\infty(-\alpha)^k\frac{(2k)!\:L(2k,-\beta)}{k!} $$ where $L(n,x)$ is the $n$-th Laguerre polynomial evaluated at $x$. In this case, I know that $...
5
votes
3answers
56 views

prove simple binomial sum, combinatorics

I want to prove that: $$\large\sum_{i = 1}^{n} \binom{n}{i}\binom{n}{i-1} = \binom{2n}{n-1}$$ On the right hand side we simply have the coefficient of $x^{n-1}$ of the term $(1+x)^{2n}$ But on the ...
14
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2answers
965 views

Math Olympiad Summation Question

Let $a_i \in \{-1,1\}$ for all $i=1,2,3,...,2014$ and $$M=\sum^{}_{1\leq i<j\leq 2014}a_{i}a_{j}.$$ Find the least possible positive value of $M$. Came across this question in a Math Olympiad and ...
1
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1answer
35 views

Summation of $A\cos (\omega n+\phi)$ [closed]

I'm trying to evaluate the following summation: My original problem is $$\lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^N \left|A \cos(\omega n+\phi)\right|^2$$ Now I'm stuck at calculating the ...
18
votes
5answers
901 views

A strange combinatorial identity [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
5
votes
1answer
66 views

Combinatorial proof of a certain alternating sum of binomial coefficients

The following identity appeared as a question earlier today $$\displaystyle\sum\limits_{k=0}^n (-1)^k\binom{m+1}{k}\binom{m+n-k}{n-k} = \begin{cases} 1\ \text{if}\ n=0 \\ 0\ \text{if}\ n>0 \end{...
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2answers
88 views

Find the properties of the sum $\sum_{k=0}^n (-1)^k\binom{m+1}{k}\binom{m+n-k}{n-k}$

I have to show that $$\displaystyle\sum\limits_{k=0}^n (-1)^k\binom{m+1}{k}\binom{m+n-k}{n-k} = \begin{cases} 1\ \text{if}\ n=0 \\ 0\ \text{if}\ n>0 \end{cases}$$ My try: I have tried to use snake ...
2
votes
2answers
43 views

arccot limit: $\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$

I have to find the limit of this sum: $$\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$$ I tried using sandwich theorem , observing: $$\cot ^{-1}(r^3)\leq\cot ^{-1}(r^2+\frac{3}{4})\leq\cot ^{-1}(r^...
0
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0answers
17 views

Jacobian matrix of summation function

So let's say I have a function like this $(\mu_{ij})_{i,j=1,...,t;i+j>t}\longmapsto \sum_{i,j;i+j>t} \mu_{ij}$ and I need to find the Jacobian matrix of that function. I tried to calculate it ...
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2answers
37 views

How to prove $\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)} $ diverges?

$$\sum_{n=1}^{\infty} \frac{3^n +7n}{2^n (n^2+1)} $$ It seems clear to me that this seires diverges since the dominant term is $(3/2)^n$, a geometric series with $r > 1$ However I am required to ...
-1
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1answer
16 views

Working through summations to show equation

Given equation 1: $$E = \sum_{k=1}^N \tau x_k g(\frac{n_k}{\tau}) + \sum_{k=1}^N n_kh(\frac{n_k}{\tau})$$ equation 2: $$E = \frac{1}{2}\gamma X^2 + \epsilon \sum_{k=1}^N |n_k| +\frac{\eta*}{\tau}\...
-1
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2answers
34 views

are these summations equal

Give Function A: $$ \frac{1}{2} \gamma X^2 - \frac{1}{2}\gamma \sum_{i=1}^N n_k^2 $$ and Function B: $$ \epsilon \sum_{i=1}^N |n_i| + \frac{\eta}{\tau} \sum_{i=1}^N {n_i}^2$$ Can you show that ...
7
votes
3answers
799 views

relationship between sum of squares and sum

I have to admit I am not good at math since it's been a while since I did the last math problem. I am working on a project where there is a problem that can be summarized like this: if $\sum_{i=1}^{n}...
4
votes
3answers
95 views

How am I miscalculating the telescoping sum $\log(\frac{n+1}{n})$?

All values of $a_n = \log(\tfrac{n+1}{n})$ must be positive since $\tfrac{n+1}{n} > 1$. Hence $\sum_{n=1}^{\infty} a_n$ must be greater than $0$. However when I calculate it as a telescoping sum, ...
2
votes
2answers
39 views

Find the sum of a non-geometric series

Find the sum of the series or show that the series is divergent. $$\sum_{n=0}^\infty \frac{5^n-2}{7^n}$$ So, I've established that this series is convergent via the comparison method; however, I'm ...
2
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1answer
19 views

Limit of a floor sum

How can i prove that $ \forall x \in \mathbb{R} \displaystyle \lim_{n \to \infty} \dfrac{\left \lfloor{x}\right \rfloor+\left \lfloor{2x}\right \rfloor+\cdots+\left \lfloor{nx}\right \rfloor}{n^2} = \...
0
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0answers
68 views

Sequence of integers in given range that sums up to given value

I'm trying to find out, if there is a way to find the total number of possible combinations of integers $x_i \in [l,u] \cap \mathbb{Z}$ for all $i = 1,\ldots,n$ that sum up to $A$. Generally, \begin{...
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2answers
42 views

Convergence of $\sum \sin\frac{(-1)^n}{n^p}$

$$\sum_{n=1}^{\infty} \sin\frac{(-1)^n}{n^p}\quad p>1$$ My attempt: $$\sum_{n=1}^{\infty} \sin\frac{(-1)^n}{n^p} = \sum_{n=1}^{\infty} (-1)^n\sin\frac{1}{n^p} $$ And $\sum_{n=1}^{\infty} \...
0
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1answer
25 views

Summation that gives perfect squares

For $n=1,2,3,4$ upto $50$. How many $s(n)$ will be perfect squares? The answer given is $3(n=1,8,49)$. What will be the approach for such questions?
2
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1answer
20 views

Check convergence and sum of a sum of finite sum.

$$\sum_{n=1}^\infty \sum_{k=1}^m \left(\frac{x_k}{y}\right)^n\quad 0<x_k<y$$ My attempt: Convergence: Since $\frac{x_k}{y} <1$ we can conclude that: $$\sum_{k=1}^m \left(\frac{x_k}{y}\...
2
votes
2answers
78 views

Asymptotic for combinatorial function

Let $$F_q(k) = \sum_{n=1}^{\infty} \binom{n-1}{k} \binom{1/2}{n} q^n$$ be a function on $\mathbb{N}$. I am interested in the asymptotic behavior of $F$. Any ideas how to tackle it?
2
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1answer
44 views

Computing a summation using Maclaurin series and infinite products

Using the Maclaurin series for $\sin z$ and $\sinh z$, as well as the infinite products $$\sin z = z\prod_{n=1}^\infty\left(1 - \frac{z^2}{n^2\pi^2}\right)$$ and $$\sinh z = z\prod_{n=1}^\infty\...
1
vote
1answer
24 views

Derivative of a variable times its summation

Say you want to calculate $$ \frac{\partial}{\partial x_i} x_i(a - b \sum_{i=1}^N x_i). $$ I assume the term $bx_i \sum_{i=1}^N x_i$ is derived using the product rule, but I am unsure what the ...
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0answers
58 views

Evaluate the combination of $\sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j}$

Can any one help me please to get the approximate result of this combination problem using asymptotic notation: $$ \sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j} $$ Thanks
0
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1answer
38 views

How to simplify $\sum_{k=0}^{\infty} \binom{2k}{k} (sp)^kq^{k}$

$\sum_{k=0}^{\infty} \binom{2k}{k} (sp)^kq^{k}$ = $\sum_{k=0}^{\infty} \binom{2k}{k} (sp)^kq^{2k - k}$ I know that if I had a truncation, ie, $\sum_{k=0}^{N} \binom{2k}{k} (sp)^kq^{k}$, I would have $...
0
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1answer
23 views

Simplified form for a Newton's sum? $a_1^k + a_2 ^k + \ldots + a_n^k = k $ for $k=1,2,\ldots,n$.

Let $ a_1, a_2, a_3, \ldots , a_n $ be complex number satisfying $ \displaystyle \sum_{j=1}^n a_j ^k= k $ where $ k =1,2,\ldots, n $. Prove (or disprove) that $\displaystyle \sum_{j=1}^n a_j ^{n+1} $...
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0answers
16 views

Least degree polynomial and Newton's sum? $a_1^k + a_2 ^k + \ldots + a_n^k = k $ for $k=2,3,\ldots,n+1$.

Let $ a_1, a_2, a_3, \ldots , a_n $ be complex number satisfying $ \displaystyle \sum_{j=1}^n a_j ^k= k $ where $ k =2,3,\ldots, n+1 $. Prove (or disprove) that the least degree polynomial with ...
1
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1answer
26 views

Why is $((n-1) \mod 9)+1$ equal to summing all digits till one digit is left?

There was a question on SO on how to, in excel, sum all digits in a number until you are left with one single digit. The correct answer, in excel format, turns out to be ...
5
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6answers
867 views

Find the sum of the infinite series $\sum n(n+1)/n!$

How do find the sum of the series till infinity? $$ \frac{2}{1!}+\frac{2+4}{2!}+\frac{2+4+6}{3!}+\frac{2+4+6+8}{4!}+\cdots$$ I know that it gets reduced to $$\sum\limits_{n=1}^∞ \frac{n(n+1)}{n!}$$ ...
0
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1answer
48 views

Writing the product $\sum\limits_{r=0}^\infty \frac{z^r}{r!} \sum\limits_{s=0}^\infty \frac{z^{-s}}{s!}$ as a power series in $z$

My lecturer states that the product $$\sum_{r=0}^\infty \frac{z^r}{r!} \sum_{s=0}^\infty \frac{z^{-s}}{s!}$$ can be written as (with $n = r-s$) $$\sum_{n=0}^\infty z^n\sum_{r=n}^\infty \frac{1}{r!(...
0
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1answer
23 views

Upper bound for $\sum_{x=1}^l \left(\frac{s}{x}\right)^x$

Let $l,s$ be some large numbers (if it helps, you might assume $s \gg l \gg 1$) and consider $$S:=\sum_{x=1}^l \left(\frac{s}{x}\right)^x.$$ What one can easily do is the following: $$S \leq \sum_{...
1
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0answers
38 views

Find the sum of a Cos series.

I'm using the following equation to generate a number series. $v =-c\ \cdot \ \cos \left(\frac{t}{d}\cdot \left(\frac{\pi }{2}\right)\right)+c\ +\ b$ Values I'm using to solve the series are: $b = ...
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0answers
43 views

limit of complicated sum that wolfram alpha cannot solve

Let $1 \leq d\leq \frac{1}{2}\log^3(n) \sqrt{n}$. We would like to show that for any such $d$ we have $$\sum_{x=0}^{\lceil \log^6(n) d \rceil}\left(\log^3(n) \sqrt{n}\right)^{d+x} e^{-\frac{\log^4(...
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2answers
60 views

Show that $\sum _{k=1} ^N \frac 1 {\sqrt {k^2 + 1} + k} > \frac 1 2 \ln \frac {2N+1} 3$, where $N$ is natural number.

Show that for $N = 1,2,3,\dots$ we have $$\sum _{k=1} ^N \frac 1 {\sqrt {k^2 + 1} + k} > \frac 1 2 \ln \frac {2N+1} 3$$ I got this as a calculus homework. I am supposed to show this, but it doesn'...
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4answers
168 views

Summation Problem (lower limit is variable) [closed]

$$\sum_{j=i}^n 2$$ I am having difficulty solving this summation. Can i have hint or solution to this problem?
0
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3answers
79 views

Let $(\sqrt{3} + \sqrt{2})^5 = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$ Find $a+b$.

Let $$(\sqrt{3} + \sqrt{2})^{\color{red}{5}} = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$$ Find $a+b$. I don't know if that's supposed to be $\color{red}{5}$ or $\color{red}{3}$. By binomial ...
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3answers
200 views

A Ramanujan-type identity: $11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$

Out of curiosity, why it is these sums yield a rational answer? $$11\sum_{n=1}^{\infty}\frac{n^3}{e^{2n\pi}-1}-16\sum_{n=1}^{\infty}\frac{n^3}{e^{4n\pi}-1}=\frac{1}{48}$$ I found this identity ...
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0answers
33 views

Solve $\sum_{n=1}^p\frac{2^ne^{nx}}{\binom{2n}n}$

Related to another question, I need to solve the following summation: $$\sum_{n=1}^p\frac{2^ne^{nx}}{\binom{2n}n}$$ Solved in terms of $x$ and $p$, and $\binom{2n}n=\frac{(2n)!}{(n!)^2}$ I could ...
0
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0answers
34 views

Need to understand this summation with max notation

Firstly, apologies needed for my math description if it does not sound right. I have come across a paper where I saw a summation notation with a max function in it which I am little confused to ...
3
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2answers
49 views

Another Hockey Stick Identity

I know this question has been asked before and has been answered here and here. I have a slightly different formulation of the Hockey Stick Identity and would like some help with a combinatorial ...
4
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3answers
104 views

Find $\frac{1}{7}+\frac{1\cdot3}{7\cdot9}+\frac{1\cdot3\cdot5}{7\cdot9\cdot11}+\cdots$ upto 20 terms

Find $S=\frac{1}{7}+\frac{1\cdot3}{7\cdot9}+\frac{1\cdot3\cdot5}{7\cdot9\cdot11}+\cdots$ upto 20 terms I first multiplied and divided $S$ with $1\cdot3\cdot5$ $$\frac{S}{15}=\frac{1}{1\cdot3\cdot5\...
3
votes
3answers
46 views

Explanation of the Sum of an Infinite Series Equation

I've been presented with the following infinite sum (where $P$ is the probability of an event, and $1-P$ is therefore the probability of it not occurring. I was given the following equation as fact: $...
1
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1answer
41 views

The analytic extension of $\sum_{k=1}^n\frac1k$ and an induction

The analytic extension of the sum of the first $n$ reciprocals is given as $$\sum_{k=1}^n\frac1k=\int_0^1\frac{x^n-1}{x-1}dx$$ I am wondering if $\frac1{n+1}+\sum_{k=1}^n\frac1k=\sum_{k=1}^{n+1}\...
1
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0answers
25 views

Limit of a certain sum

I need to show that $$\sum_{i=0}^{m} \binom{m}{m-i}\binom{m^2-m}{i} (1-p)^{\binom{i}{2} + i m} \bigg/ \binom{m^2}{m} (1-p)^{\binom{m}{2}} \to 0$$ as $m \to \infty$, where $p = \frac{1}{m}$, and the ...
0
votes
0answers
35 views

Calculating infinite series for a hospital waiting queue

For my project, I had to simulate a hospital waiting queue, and ended up stuck with this equation. $$ 1=\sum_{i=0}^\infty \left(\frac{\lambda}{\mu+i\gamma}\right)^iP_0 $$ Could any kind soul help ...
0
votes
0answers
63 views

Prove with induction that $\sum_{k=0}^{n-1}x^{k}=\frac{x^n-1}{x-1}$

Prove with induction that $$\sum_{k=0}^{n-1}x^{k}=\frac{x^n-1}{x-1}$$ It seems simple but I have tried for I don't know how long by now... Anyone can manage this?
0
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0answers
10 views

Can you use Dirichlet's hyperbola method with any of these pathological logarithms?

I would like to learn Dirichlet's hyperbola method in some of myself next posts. I know its meaning and relationship with the divisor function and lattice problems, but in this ocassion I want to ...
3
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5answers
66 views

How to prove that $\sum_{i=j}^nn-i = \sum_{i=1}^{n-j}i$?

Trying to solve question 2-3 from Skiena's Algorithm Design Manual which asks to find the runtime of the following loop: ...
13
votes
6answers
211 views

How to prove that $\sum_{i=0}^n 2^i\binom{2n-i}{n} = 4^n$.

So I've been struggling with this sum for some time and I just can't figure it out. I tried proving by induction that if the sum above is a $S_n$ then $S_{n+1} = 4S_n$, but I didn't really succeed so ...