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

learn more… | top users | synonyms

2
votes
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 ...
2
votes
2answers
75 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
votes
1answer
42 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 = ...
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 ...
1
vote
0answers
47 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
votes
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
votes
1answer
22 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} ...
1
vote
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 ...
-3
votes
0answers
19 views

Triple sum on Maple

I'm new to Maple and having a problem indexing a triple sum. It's a solution for an EDP and it's like the first image.: $\eta$, $\mu$, and $\beta$ are eigenvalues, but just $\mu$ is transcendental ...
1
vote
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 ...
4
votes
6answers
837 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
votes
1answer
47 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 ...
0
votes
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 ...
1
vote
0answers
37 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 = ...
0
votes
0answers
42 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} ...
0
votes
2answers
57 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 ...
0
votes
4answers
163 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?
-1
votes
3answers
71 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 ...
8
votes
3answers
194 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 ...
0
votes
0answers
32 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
votes
0answers
33 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
votes
2answers
48 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
votes
3answers
100 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 $1\cdot3\cdot5$ ...
2
votes
3answers
35 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
vote
1answer
40 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 ...
1
vote
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
34 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
votes
0answers
7 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
votes
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
208 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 ...
1
vote
3answers
703 views

How to prove the sum of combination is equal to $2^n - 1$

One of the algorithm I learnt involve these steps: $1$. define a set $S$ of $n$ elements $2$. form a subset $S'$ of $k$ choice from $n$ elements of the set $S$ ($k$ starts with $1$), which is ...
0
votes
2answers
56 views

Looking for a closed form for $a_m =\sum_{k=0}^{\infty}\binom{k+m}{m}\frac{1}{4^{k}(2(k+m))!}$

I have this sequence $$ a_m =\sum_{k=0}^{\infty}\binom{k+m}{m}\frac{1}{4^{k}(2(k+m))!} $$ and there seems to exist a patern arising when it is evaluated by WA. It involves $\cosh(1/2)$ and ...
0
votes
2answers
35 views

Value of n summations of 1 $\sum_{0 \le a_1\lt a_2 \lt a_3 …\lt a_k \le n}1$

I need to find $$\sum_{0 \le a_1\lt a_2 \lt a_3 ...\lt a_k \le n}1$$ My attempt: I think it is equal to $ ^nC_k $ as $$\sum_1^n{1} = n = ^nC_1$$ $${\sum \sum }_{0\le i \lt j \le n} 1 = \frac ...
1
vote
3answers
31 views

Evaluate $\int_1^N \frac{-3N+6t-3}{t^3(N-t+1)^4}dt$ when $N=3$ or $N=5$

Let the Cauchy product $$(\zeta(3))^2=\sum_{n=1}^\infty c_n,$$ where $$c_n=\sum_{k=1}^n\frac{1}{k^3(n-k+1)^3},$$ and $\zeta(3)$ is the Apèry constant. Taking $f(x)=\frac{1}{x^3(N-x+1)^3}$ in Abel's ...
1
vote
3answers
63 views

Matlab sum is wrong: Double symsum gives incorrect result

I'm trying to calculate the double sum $$ \frac{1}{10} \sum_{x=1}^{10} \left( \frac{1}{x} \sum_{n=0}^{floor(log_{10}x)} 10^n \right).$$ In MATLAB, my result is ...
1
vote
1answer
21 views

Finding $\sum_{j=m}^{n}\frac{a^{j-m}}{N-j}$

How can we tackle $$\sum_{j=m}^{n}\frac{a^{j-m}}{N-j}$$ when $0<m<n<N$. I have been using Euler-Maclaurin sum to change this object to some integral but it gets a bit messy. I appreciate any ...
1
vote
0answers
28 views

Series with Markov Chains Probabilities

Notation For each $t \in \mathbb{N}$, let $h_t \in H$ be a random variable that follows a Markov chain, and $h^t \equiv \{h_0,h_1,\dots,h_t\} \in H^t$. Let $\Pi(h^{t})$ be the probability that a ...
0
votes
1answer
27 views

How can you simplify this expression for amount of triangles?

I was given a question as a challenge were I was supposed to find a formula to find how many triangles there are when you draw $n$ and $m$ amount of lines from points $N$ and $M$ to the opposite ...
1
vote
1answer
27 views

Do not understand algebra technique used to computer summation

I am going through a practice exam for my Discrete Mathematics class and do not understand the algebra used in the following summation computation. Summation to compute: Answer: What I don't ...
0
votes
1answer
25 views

Limit of a series with a lot of dependencies

Let $n \rightarrow \infty$ and consider $$\sum_{x=\lfloor \log^6(n)\rfloor}^{\lceil \frac{n}{\log^2(n)}\rceil} \left(\frac{n}{\log^2(n) x} e^{-\frac{\log^{16}(n)}{n}}\right)^x$$ Do we know anything ...
0
votes
2answers
27 views

Help finishing proof for $\sum_{i=1}^{n-1} (-1)^{i+1}i! \leq \frac{(2n)!}{2}$

I need help finishing this proof. I've come to a point where I don't know how to continue. I need to prove that the following inequality is true for all positive integers $n$. $$\sum_{i=1}^{n-1} ...
4
votes
0answers
56 views

Calculating $\lim_{n \to \infty}\left(\dfrac{1^n +2^n +3^n + \ldots + n^n}{n^n}\right)$ [duplicate]

Evaluate : $$\lim_{n \to \infty}\left(\dfrac{1^n +2^n +3^n + \ldots + n^n}{n^n}\right)$$ I tried using squeeze theorem, but I couldn't find the proper inequality. I also thought of ...
0
votes
0answers
24 views

Compute limit of a double-sum

Let $n \rightarrow \infty$. I would like to compute the limit $$\sum_{x,y=1/2 \log^c(n)}^{\frac{n}{log^{d}(n)}} e^{(x+y)\log n}e^{-C x y \log^c(n)}$$ where $c,d,C \in \mathbb{N}$ can be chosen ...
0
votes
3answers
59 views

Request for a proof that $\sum\limits_{i=1}^n i^{k+1}=(n+1)\sum\limits_{i=1}^n i^k-\sum\limits_{p=1}^n\sum\limits_{i=1}^p i^k$

Prove $$\sum_{i=1}^n i^{k+1}=(n+1)\sum_{i=1}^n i^k-\sum_{p=1}^n\sum_{i=1}^p i^k \tag1$$ for every integer $k\ge0$. By principle of induction, $$\sum_{i=1}^n i = n(n+1)- \sum_{p=1}^n p$$ ...
0
votes
1answer
16 views

double summation notation

In a paper I am studying, the author writes $$\sum_{{i=1}\atop {k=1}}^{N+1} C_i \eta_k$$ How are the two indices to be interpreted? In other words, how would this expression be written using sigma ...
0
votes
1answer
52 views

How to prove this quasi-geometric trigonometric series identity without induction

$$\frac{2}{\sin{x}}\sum_{r=1}^{n-1} \sin{rx}\cos{[(n-r)y]} \equiv \frac{\cos{(nx)}-\cos{(ny)}}{\cos{x}-\cos{y}} - \frac{\sin{(nx)}}{\sin{x}}$$ The identity can be tediously proven using the Axiom of ...
1
vote
1answer
20 views

On computations around $\sum_{n=1}^N\frac{n\Lambda(n)}{n+N}$, where $\Lambda(n)$ is von Mangoldt function

By specialization with $F(x)=\frac{1}{1+x}$ in Apostol's Theorem 4.17 (Apostol, Introduction to Analytic Number Theory (Springer)), for intergers $N\geq 1$ one has $$\frac{\log ...
1
vote
1answer
31 views

Sigma notation for following nested loop

How may the following programming statement be written as summation? ...
0
votes
1answer
40 views

Extending binomial identity $ \sum\limits_{k=0}^n\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$ to $0<x<1$

I found in Matlab that $$ \sum_{k=0}^n~\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$$ for $1\leq x< n$ only (I am about 95% sure of this since the sum is numerically unstable and cannot give ...