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

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1
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1answer
46 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
412 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
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10answers
160 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
52 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
106 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
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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
182 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
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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
47 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
70 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} \...
48
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
262 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?
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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
votes
1answer
53 views

A binomial sum identity

Let \begin{align*} f(n, r, \pi, k) &= \sum_{z=0}^{n}\sum_{s=0}^{r}\binom{z}{s}\binom{n}{z}\binom{n-z}{r-s}(-1)^{r+s}\left(\frac{\pi}{1-\pi}\right)^{r/2-s}\pi^{z}(1-\pi)^{n-z}z^k \end{align*} I am ...
1
vote
2answers
31 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
56 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
39 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
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0answers
31 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
85 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
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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
57 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
59 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
716 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
30 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
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3answers
57 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
971 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 ...
19
votes
5answers
920 views

A strange combinatorial identity: $\sum\limits_{j=1}^k(-1)^{k-j}j^k\binom{k}{j}=k!$ [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{...
1
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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
45 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
votes
0answers
20 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 ...
1
vote
3answers
48 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
votes
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
votes
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
806 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
42 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
votes
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
votes
2answers
43 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} \...