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

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2
votes
1answer
45 views

Evaluation of Infinite series summation.

For any Positive integer $n\;,$ Let $t(n)$ denote the integer closest to $\sqrt{n}\;,$ Then value of $\displaystyle \sum^{\infty}_{n=1}\frac{2^{t(n)}+2^{-t(n)}}{2^n}$ $\bf{My\; Try::}$ Here ...
3
votes
2answers
64 views

How to find the summation of this infinite series: $\sum_{k=1}^{\infty} \frac{1}{(k+1)(k-1)!}(1 - \frac{2}{k})$

I've been trying to figure out the following sum for a while now: $$\sum_{k=1}^{\infty} \frac{1}{(k+1)(k-1)!}\left(1 - \frac{2}{k}\right)$$ I'm pretty sure that this doesn't evaluate to $0$. ...
0
votes
0answers
18 views

extracting a function out of an equation

I encountered the following problem in my thesis. We have an equation as follows: $\phi(s)=\sum^\infty_{n=1}P(n)\int^{\infty}_{0}e^{-st}f(t|n)dt=\sum^{\infty}_{n=1}[(1-q)M_1(s)]^{n-1}qM_2(s)$ in ...
0
votes
1answer
14 views

Value of the series $\sum_{n=1}^{\infty} H_n^{(2)}x^n$ and $\sum_{n=1}^{\infty} H_n^{(3)}x^n$

I can find series $$\sum_{n=1}^{\infty} H_nx^n = -\frac{\log(1-x)}{1-x}$$ but I can't find $$\sum_{n=1}^{\infty} H_n^{(2)}x^n$$ and $$\sum_{n=1}^{\infty} H_n^{(3)}x^n$$ $H_n^{(p)}=1+\frac{1}{2^p}+...
1
vote
0answers
24 views

Finding a closed mathematical form for a parametrized function series

I am dealing with the following parametrized function series defined by $$ F(x) := \sum_{n=0}^{\infty} 2\left( \varphi_n(x)-(n+1)(n+2)x^2 \psi_n(x) \right)x^{n+1} \, , $$ where $x \in [0,1)$. The ...
2
votes
1answer
62 views

Value of the series $\sum_{n=1}^{\infty}\big (\frac{H_n}{\binom{3n}{n}}\big)^2$

I want to find the value of following series $$ \sum_{n=1}^{\infty} \left(\frac{H_n}{\binom{3n}{n}}\right)^2\tag{1} $$ where $H_n = 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$. I know that $$\sum_{...
6
votes
0answers
108 views

Help with the following summation when $x^{37}=1,x\neq 1$

I want to find the following summation $\text{Let }x^{37} = 1 \text{ and } x \neq 1,$ $\\ \text{Find the summation of }$ $$\frac{1}{(1+x+x^2+x^3)^3}+\frac{2}{(1+x^2+x^4+x^6)^3}+...+\frac{36}{(1+x^{...
1
vote
1answer
31 views

How sum of exponential variables is a gamma variable [duplicate]

I have the task to calculate $P(S_{100}\geq 200)$ where $S_{100}=\sum^{100}_{i=1} X_i$ and $X_i$, $i=1,2, \cdots, 100$ are independent $exp(\lambda)$ random variables. One method is to use the fact ...
2
votes
3answers
51 views

Summation of a series with 2 different methods gives 2 different answers

The objective here is to find the value of $S$, where $S$ is given by, $$S = 1-{1\over2}+{1\over3}-{1\over4}+...$$ I did this using two methods, but both the methods give different answers. Method 1: ...
3
votes
1answer
105 views

What should $\int \frac{1}{x} dx$ equal to?

Before you say that $\int \frac{1}{x} dx$ is equal to $\ln|x| +C$ due to positve and negative, I would like to show you why it is not convincing to me. Problem 1 and its possible solution. \begin{...
0
votes
1answer
28 views

a[i] denote number of friends i-th student has. c[j] denote frequency having at least j friends. Show that: ∑a[i]=∑c[j]. [duplicate]

Q. A class has 100 students. Let a[i], 1≤i≤100, denote the number of friends the i-th student has in the class. For each 0≤j≤99, let c[j] denote the number of students having at least j friends. Show ...
0
votes
5answers
428 views

Mathematical induction using Sigma [closed]

I have attached an image of a kind of mathematical induction question that i have never seen before. I attached it because i don't know how to type all the symbols out properly, i'm sorry again would ...
1
vote
1answer
45 views

Relation between $\gcd$ and Euler's totient function .

How to show that $$\gcd(a,b)=\sum_{k\mid a\text{ and }k\mid b}\varphi(k).$$ $\varphi$ is the Euler's totient function. I was trying to prove the number of homomorphisms from a cyclic group of order ...
8
votes
3answers
407 views

Combinatorial formulas and interpretations

I found that $$ \sum_{j=0}^{s}(n-s+j)!\binom{s}{j}(s-j)! =s! \sum_{j=0}^{s} \frac{(n-s+j)!}{j!} = \frac{(n+1)!}{n+1-s}$$ I proved this formula with induction, but I was wondering if there is a (...
1
vote
10answers
157 views

Find the sum of the squares of the first $n$ natural numbers

I've been asked to find the sum of the squares of the first $n$ natural numbers. My initial thought was to just program a brute-force solution but I was wondering if there is a mathematical formula ...
2
votes
1answer
51 views

How can I prove that $(a + b )\oplus(a + c)$ is not possible to simplify. Or is it?

I was trying to simplify the following expression $(a + b )\oplus(a + c)$, where $+$ is just a simple addition of two numbers and $\oplus$ is a binary xor operation. By simplifying I mean exanding or ...
0
votes
0answers
14 views

Boundary of $\sum_{j}x_j(x_j-x_i)$ for $x_i \in[0,1]$

Does $\sum_{j}x_j(x_j-x_i)$ for $x_i\in[0,1]$ and $0\le i,j\le N-1$ have a upper and lower boundary? And how to calculate them? Thanks!
0
votes
3answers
90 views

If $\omega = e^{(\frac{2\pi i}{n})}$ why $1+ \omega + \omega^{2} + … + \omega^{n-1} = 0 $? [duplicate]

Let $\omega = e^{(\frac{2\pi i}{n})}$ why $1+ \omega + \omega^{2} + ... + \omega^{n-1} = 0 $? I saw this on a algebra PPT slice. However the teacher did not explain why this equation is correct, can ...
0
votes
0answers
37 views

How to calculate $\sum_{n=1}^\infty {p^{n-1}}{(1-p)^{n}} \frac{1}{n} {2n-2 \choose n-1} $?

How to prove $\sum_{n=1}^\infty {p^{n-1}}{(1-p)^{n}} \frac{1}{n} {2n-2 \choose n-1} $ is 1 if $0 \le p \le \frac{1}{2}$ and smaller than 1 if $\frac{1}{2} \lt p \le 1$? I came up with this ...
10
votes
3answers
171 views

Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$

Numerical calculations suggest that $$\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,\int_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx=1.553767373413083673460727...$$ How can we prove ...
0
votes
2answers
39 views

Proving, that $\text{Arg}(-i\sin(x))=\pi/2\text{sgn}(x)$ on $(-\pi,\pi)$

Alright. I thought, that $\text{Arg}(-i\sin(x))=3\pi/2$, however, the Wolfram Alpha tells a different story. I am sure that it must be kind of true, because $\text{Arg}(\sin(x))$ is the result of sum ...
1
vote
0answers
46 views

Evaluating convergence of a non-trivial series

I am trying to evaluate whether the following limit is finite (as opposed to being $\infty$): $$\lim_{n \to \infty} \sum_{k=2}^n \frac{1}{n-1} \left \{\sum_{i=2}^l (i-1)\frac{(n-i)!}{(n-i-k+2)!} \...
-2
votes
3answers
68 views

How is this series rearranged?

I'm stuck at this. How is RHS rearranged? Is it a change of index? $$ \sum_{n=1}^{2N} \frac{1}{n} - \sum_{n=1}^{N} \frac{1}{n} = \sum_{n=N+1}^{2N} \frac{1}{n} $$ I'm stuck here: $$ \sum_{n=1}^{2N} \...
47
votes
16answers
4k views

How can you prove that $1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$ without using induction?

Using mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result ...
0
votes
1answer
33 views

Evaluate limit of a sum that includes summed term

I am trying to determine whether the limit $$ \lim_{n \to \infty}\sum_{k = 2}^{n}\left(\frac{n - k}{n - 2}\right)^{2k} \left(\frac{l - 1}{2}\right)^{k} $$ exists and is finite. No idea how to ...
0
votes
0answers
11 views

Upper bound for a graph related finite sum

At the moment I am looking into undirected graphs $G=(V,E)$ with node set $V=\{1,\ldots,M\}$ and edge set $E$. We can assume that they are connected by the way. Lets denote edges from $i$ to $j$ by $(...
4
votes
1answer
54 views

Sum of all sine harmonics

I was discussing this with my calculus teacher, but she didn't come up with anything. I would like to take an infinite sum of functions (sine specifically) but don't know how to do that. I would ...
5
votes
2answers
237 views

Closed form for $1^k + … + n^k$ (generalized Harmonic number)

This question must have been asked, it's just very hard to search for such questions. I'm looking for the cleanest method I can find for getting a closed form formula for $\sum_{i=1}^n i^k$ ...
7
votes
1answer
147 views

Prove that sum is convergent

How to prove that the following sum is convergent? $$\sum_1^\infty\frac{\sin(n + \ln{n})}{n}$$ I tried to use formula $$\sin(n+ \ln{n}) = \sin{n}\cos \ln{n} + \sin \ln{n}\cos{n}$$ and $$\sum_1^N \sin{...
1
vote
0answers
28 views

Evaluating the limit of a sum when the variable being summed to is in the sum

I want to evaluate the limit $\displaystyle{\lim_{n \to \infty} \sum_{k = 2}^{n}\frac{n!}{\left(n - k + 1\right)!}\,\frac{1}{n^{k}}\,c^{k}}$ Any ideas on how to do this?
-1
votes
2answers
54 views

Asymptotics of $\sum_{n}e^{-n^{2}}$.

Define the function $S(N)$ as $$S(N)=\sum_{n=0}^{N}e^{-n^{2}}$$ I am interested in the asymptotic behavior of $S(N)$ for large $N$. It is clear by the ratio test that $\lim_{N\rightarrow\infty}S(N)$ ...
1
vote
2answers
30 views

Can anyone help me with this finite sum?

I have to calculate the sum $\displaystyle\sum_{k=1}^n \displaystyle\frac{3^k}{3^{2k+1}-3^k-3^{k+1}+1}$ We can re-write the sum as follows $\displaystyle\sum_{k=1}^n \displaystyle\frac{3^k-1+1}{(3^{...
2
votes
0answers
53 views

Ideas on how to simplify or approximate this nasty sum

I have a sum (let's call it $p$): $$p:= \frac{1}{n!}\sum_{i=2}^l (i-1)\frac{(n-k)!}{(n-k-i+2)!}(n-i)!$$ where $l, n, k$ are fixed positive integers, and $k \leq n$. I'd like to either simplify or ...
2
votes
0answers
36 views

Name of dominated convergence for sums

Having a sequence $(a_n(j))_{n}$ where every element of the sequence also depends on $j\in\mathbb{N}$. If $\sum_{n=1}^\infty \sup_{j\in\mathbb{N}} |a_n(j)| < \infty$, then the following (assuming ...
1
vote
0answers
23 views

Infinite sum of inverse trigonometric function [closed]

$$\sum_{n=0}^{\infty} atan((acot(n^2+n+3))/(1+acot(n+1)acot(n+2)))$$
4
votes
3answers
83 views

What is $\lim_{x\to \infty} 2\sqrt{x}- \sum_{n=1}^x {1\over \sqrt{n}}$? [duplicate]

I ask this because I noticed the partial sum $\sum_{n=1}^x {1\over \sqrt{n}}$ is very close to $2\sqrt{x}$, so close in fact that it appears their difference approaches a constant value, like $H_x$ ...
3
votes
1answer
45 views

summation of $\sum_{k=0}^{\infty}x^{n^{k}}$

Let $x\in (0,1)$ and $n\in 2\mathbb{N}+1$ be fixed. the series $$\sum_{k=0}^{\infty}{x^{n^{k}}}$$ is convergent by Ratio Test. what is the sum of the series ?
0
votes
0answers
24 views

summation of $\sum_{k=0}^{\infty}{q^{\sum_{i=0}^{k}{n^{i}}}}$

Let $q\in (0,1)$ be fixed. Consider the sequence $\{q^{\sum_{i=0}^{k}{n^{i}}}\}_{k=0}^{\infty}$, where $n$ is a fixed odd positive integer. This sequence is convergent to zero by dini's theorem. set $$...
0
votes
0answers
31 views

Power Quantum Series And It's Sum.

Let $I$ be the interval $(-\theta, \theta), \theta=q^{\frac{1}{1- n}}$, $n\in 2\mathbb{N}+1$ and $q\in (0,1)$ are fixed. Define a function $h(t):=qt^{n}$. One can see that the $k$-th order iteration ...
0
votes
1answer
33 views

The reason of $\int_{-\infty}^{\infty}\mu_k^2(x)dx=1$

Is there anyone could tell me why if $$\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}}=\int_{-\infty}^{\infty} (\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}} \mu_k^2(x))dx= \sum_{k \geq 0} e^{it \sqrt{-\lambda_k}}...
2
votes
3answers
56 views

Finding the limit to infinity of a summation

The following problem was featured as a challenge in a previous exam paper and has left me stumped. Compute the following limit: $$ \lim_{n \to \infty} \frac{1}{n^{2013}} \sum_{k=1}^n k^{2012} $$ ...
4
votes
1answer
58 views

Every even integer $n>2$ is a semiprime or sum of two semiprime numbers.

Progress: A slightly stronger version of the original assumption is this: Every even integer $n>2$ is a semiprime or sum of two even semiprime numbers. I was wondering as to how this ...
6
votes
6answers
711 views

Solve summation expression

For a probability problem, I ended up with the following expression $$\sum_{k=0}^nk\ \binom{n}{k}\left(\frac{2}{3}\right)^{n-k}\left(\frac{1}{3}\right)^k$$ Using Mathematica I've found that the result ...
0
votes
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
vote
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
votes
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
vote
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{...