Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [summation]

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

2
votes
0answers
73 views

Find the limit $‎{\lim\limits_{n\to\infty}}(x \log(n^2+a^2) + \sum\limits_{k=1}^n \log(k^2+a^2) -‎ ‎\log((k+x)^2+a^2))‎$‎

I've been stuck with calculating the limit of the following problem. Can you help? $‎‎\displaystyle{\lim_{n\to\infty}}(x \log(n^2+a^2) + \sum_{k=1}^n\log(k^2+a^2) -‎ ‎‎\log((k+x)^2+a^2))‎$,‎ for $a&...
2
votes
0answers
67 views

Inequality with two sums

We have a fixed integer $k$. For two sets of non-negative numbers, $0 \leq a_i \leq M$, and $0 \leq b_i \leq M$, $i=0,1,\dotsc,k$, it is known that $$ a_0 + \frac{a_1}{n} + \frac{a_2}{n^2} + \dotsb + \...
2
votes
0answers
37 views

A Summation from A Generalized Negative Binomial Distribution

I am reading Jain and Consul "1971A Generalized Negative Binomial Distribution". The key identity of this generalised negative binomial distribution is (slightly different version): $$(1-\alpha )^{-n}...
2
votes
0answers
34 views

Binomial sum for an arbitrary function

I'm looking for some known results for sum of this type but I can't find anything. The sum is defined as: $$S(x,a,b,n)=\sum_{k=0}^n \binom{n}{k} (-1)^{k} f((a(n-k)+bk)x)$$ where $f$ is an arbitrary ...
2
votes
0answers
48 views

Closed form for a series involving the $\Gamma$ and $\zeta$ functions

I was just wondering wether one can derive a closed form for $$\sum_{n=1}^{\infty}\frac{1}{\Gamma\left(\frac{1}{n}\right)\zeta\left(1+\frac{1}{n}\right)}$$ Numerical simulation gives $S=1.20154...$ ...
2
votes
0answers
64 views

Combining inequalities to not have coefficients

We have in input an inequality cons and a set of inequalities C and we want to find a way to sum them and simplify in a way that no variable has a coefficient and that the number of constraints that ...
2
votes
0answers
48 views

mathematical notation for the sum of all the elements of the individual diagonals above the secondary diagonal.

What is the notation of the sum of the elements above the secondary diagonal for a generic nxn square matrix? I do not mean the sum of all the elements above the secondary diagonal, but rather the ...
2
votes
0answers
46 views

Finite sum involving trigonometric functions

The temperature Green's function in momentum-frequency representation of a system of free phonons (in one-dimension) is given by [1]: \begin{equation} D^{(0)}(k_n,\omega_{n'}) = -m\frac{\omega_{k_n}^...
2
votes
0answers
74 views

Simplifying Likelihood Ratio

(Answered here: https://stats.stackexchange.com/questions/372040/rejection-region-for-likelihood-ratio-test) I have a data set $((Y_1,x_1),(Y_2,x_2),...,(Y_n,x_n))$ where $Y_i$ is distributed as $N(\...
2
votes
0answers
55 views

Simplify a weighted average with enumerations as weights

Could you help me simplify the following expression: $$\forall n_0 \ge 0, \forall m_1,m_2 \gt 0, \forall i_1 \in [[0,n_0m_1]],\forall i_2 \in [[0,n_0m_2]],$$ $$m_{n_0,m_1,m_2}(i_1,i_2) = \frac{\sum_{k=...
2
votes
0answers
81 views

Closed form for $\sum_{m\geq 2, n\geq 2}\frac1{m^n-1}$

Is there a nice closed form for $\displaystyle \sum_{\substack{m\geq 2\\ n\geq 2}}\frac1{m^n-1}$ ? It's easy to prove that $\displaystyle \sum_{\substack{m\geq 2\\ n\geq 2}}\frac1{m^n} = 1$ by ...
2
votes
0answers
50 views

A sum that includes power of binomials: Possible limit?

I have the following sum: $$ \sum_{k=1}^{V} (-1)^{k-1} \frac{{V \choose k}^A}{{DV \choose k}^{A-1}} $$ Is there an approximation available for this sum? I computed this sum with python for different ...
2
votes
0answers
187 views

Sum of product of binomial coefficients & powers: closed-form?

The following sum has emerged in my research: $ \sum_{j=k}^{n}\frac{1}{2^j}{j\choose k}{n+k+2\choose n-j}; $ I am looking for a closed form for this sum (without hypergeometric series). I have ...
2
votes
0answers
62 views

How do we simplify a multiple sum that involves determinants?

Let $d \in 2{\mathbb N}$ be an even positive integer, let $p\in {\mathbb N}$ be another positive integer and let $T$ be a parameter. Let $\vec{q} := \left( q_\xi \right)_{\xi=0}^{d/2-1} \in {\mathbb ...
2
votes
0answers
93 views

Is there a simple formula for this polynomial sum?

The following seemingly arbitrary polynomial has popped up in my research on period polynomials of modular forms, and I would like to try to understand it a little better. I hope it is OK to ask about ...
2
votes
0answers
49 views

How to show following converges to $1$ uniformly?

Define $U_N=\left(\ \dfrac{\sin(\pi z)}{\pi } \right)\ \left(\ z^{-1}+\sum_{n=-N}^{n=N}(-1 )^n \left(\ (z-n)^{-1}+n^{-1} \right)\ \right)$ . where sum runs when $n$ is not equal to $0$. I want to ...
2
votes
0answers
48 views

Trouble with infinite summation involving $\sin$ function

How can I proceed for this: $$\sum_{n=-\infty}^{+\infty}{\bigg(\frac{\sin{(an+b)}}{an+b}\bigg)^2}$$ I know the answer is $\frac{\pi}{a}$, but how? I think it could be related to the Shannon sampling ...
2
votes
0answers
118 views

How to determine whether the sum ${\sum}_{k=1}^{\infty} {\sin(2^k)\over k}$ converges?

I saw a question on quora asking whether or not the sum ${\sum}_{k=0}^{\infty}{sin(2^k)\over n}$ is convergent. My opinion, and that of the other answers, is that Dirichlet's test could be used with {...
2
votes
0answers
91 views

Are there infinite many pairs of primes $(p/q)$ such that $p^3+q^3$ is a perfect square?

The pairs of primes $(p,q)$ with $p\le q\le 10^5$ such that $p^3+q^3$ is a perfect square , are (the last entry is the maximum possible exponent of the perfect power $p^3+q^3$ which is only different ...
2
votes
0answers
35 views

Upper bound on $\frac{v_{1}}{1-c_{1}x}+\cdots+\frac{v_{n}}{1-c_{n}x}$

Let $v_{1},\ldots,v_{n}$ and $c_{1},\ldots,c_{n}$ be real numbers such that $v_{i}=2\alpha_{i}^{2}$ and $c_{i}=2\alpha_{i}$ for some $\alpha\ge 0$. My question is the following: Can I get, for $x\ge 0$...
2
votes
0answers
113 views

How to prove that an alternate sum is positive? $S(p,n)=\sum _{k=0}^{n+3} (-1)^k \binom{n+2 p+4-k}{2 p+1} \binom{2 n+2 p+7}{k} (n+p+3-k)^{2 n+4}$

I'm working with this sum: $$S(p,n)=\sum _{k=0}^{n+3} (-1)^k \binom{n+2 p+4-k}{2 p+1} \binom{2 n+2 p+7}{k} (n+p+3-k)^{2 n+4}$$ and I want to prove that it is eventually positive as a function of $n$, ...
2
votes
0answers
60 views

Lower bound for $g(n)$, the number of decompositions of 2n into ordered sums of two odd primes

I was coding an algorithm that calculates $g(n)$, the number of decomposition of 2n into ordered sums of two odd primes (A002372), or the number of Goldbach partitions. I noticed i can express the ...
2
votes
0answers
49 views

Finite Sum of Power of Cosine Function

Suppose I have the following sum: Let $n=qr$ $$S_n=\sum_{j=0}^{r-1}(-1)^{qj}\cos^n{\left(\frac{j\pi}{r}\right)}$$ I am interested in which values of $r>1$ produce an integer $k$ for all $n>0$....
2
votes
0answers
39 views

On sums over non-symmetric sets

Definition: We say that a subset $A\subset \mathbb{N}_0\times \mathbb{N}_0$ is symmetric if it satisfies the property: $(a,b)\in A \Rightarrow (b,a)\in A.$ We say that the subset $A$ is non-symmetric ...
2
votes
0answers
54 views

Calculating the summation $\sum_{k=-\infty}^{\infty}\frac{e^{ika}}{1+|k|^r}$

I want to calculate $$\sum_{k=-\infty}^{\infty}\frac{e^{ika}}{1+|k|^r}$$ for $r\geq 2$ and $r\in N$. \begin{align*}\sum_{k=-\infty}^{\infty}\frac{e^{ika}}{1+|k|^r}&=1+\sum_{k=1}^{\infty}\frac{e^{...
2
votes
0answers
63 views

$\sum_{n=0}^{\infty}\frac{\sin(2n+1)x}{2n+1}$

I need to find $$\sum_{n=0}^{\infty}\frac{\sin(2n+1)x}{2n+1}$$ I know that if I find a appropriate function and expand it as a Fouries series, I can find the sum. But it seems to me like way with a ...
2
votes
0answers
22 views

Can we find Convolution Sum Boundaries without graphical plot (Analytic Method)?

I was trying to convolve two discrete sequences $x[n] = (-3)^n u(-n-1)$ with $h(n)=(\frac{1}{2})^n u[n+2] $ and was wondering if the work can be completed without having to plot the graph of the ...
2
votes
0answers
43 views

Sum involving Kaiser windows and trigonometric function

I have a sum ${\phi _1}$ where $w(n)$ is Kaiser window (centered at zero)), and ${T_s}$ is a sampling time. $${\phi _1} = \sum_{n = - N/2}^{N/2} {w{{(n)}^2}\cos } \left( {4\pi fn{T_s}} \right)$$ ...
2
votes
0answers
14 views

Discrete Fourier Transform: Given input signals, is there a way to compute the real part of DFT output without writing out the summation?

If I am given the inputs $u = [2,1,0,1,2,1,0,1]$ and asked to compute the real parts of the DFT outputs $U(m)$ I would need to apply the formula $$U(m) = \sum_{k = 0}^{N-1} u(k) e^{\frac{-j2 \pi mk}{N}...
2
votes
0answers
42 views

Which cycles are possible by repeated summation of the cubes of the digits of a number?

Here : Digital root with squared digits the possible cycles of repeated summation of the squares of the digits of a number are mentioned. What about cubes ? $1$ , $153$ , $370$ , $371$ and $407$ ...
2
votes
0answers
61 views

Calculate the sum $ \sum\limits_{i=1}^n \sum\limits_{j=1}^n a_i \mod a_j $ in $\mathbb{Z}.$

Calculate the sum $$ \sum_{i=1}^n \sum_{j=1}^n a_i \mod a_j $$ for some integer $a_i \in \mathbb{Z}.$ If we calculated the usual sum $$ \sum_{i=1}^n \sum_{j=1}^n a_i a_j $$ then it would be easy ...
2
votes
0answers
73 views

“Weighted” Summation

Since already two utents have misunderstood the question, I'm pointing it out: I've already proved the first problem, I'm using it just as an example and as motivation to the question, which is the ...
2
votes
0answers
85 views

Rewrite binomial sum with trig functions

From the multisections of sums section in wiki page on binomials, I found the following identity where for $t, s$, $0 \le t \lt s$ $$\tag{Ramus' identity} \sum_{k}{\binom{n}{t + ks}} = \frac{1}{s} \...
2
votes
0answers
85 views

Weird double summation indexing

I was reading this paper where I came across a function with a weird double summation notation. The function roughly looks like this, $$2\sum_{x = 1}^{11}(30-x)\sum_{t = 1}^{x-1} t^2$$ My ...
2
votes
0answers
48 views

Simplify $\prod_{k=1}^{l} \sum_{r=d}^m {{m}\choose{r}} \left(N-k \right)^{r} k^{m-r+1}$

Is it possible to simplify the following product of sums \begin{align} \prod_{k=1}^{l} \sum_{r=d}^m {{m}\choose{r}} \left(N-k \right)^{r} k^{m-r+1} \end{align} where $N>l$? I was thinking that ...
2
votes
0answers
48 views

Representing positive rationals as a finite sum of distinct rationals with numerator 1

A question I have been thinking about is: for every positive rational $p/q$, does there exist distinct positive integers $a_1, a_2, \dots, a_n$ (for some $n \geq 2$) such that $$\frac{p}{q} = \sum_{i =...
2
votes
0answers
65 views

Linear algebra and unknown sum limits

I have been struggling with the this one. Let's assume that: $N$ is a large integer $\epsilon$ is real and $0<\epsilon<1$ $\rho$ is real, $0<\rho<1$ and $\rho N$ is an integer $\alpha_i$ ...
2
votes
0answers
65 views

There are $N+1$ urns, in $i$th urn there are $i$ white balls and $N-i$ red balls, $n$ draws from a rand urn gave red balls, prob next ball also red?

The task: There are $N+1$ urns. In the $i$th urn there are $i$ white balls and $N-i$ red balls for $i=0,\ldots,N$. We choose a random urn and then we choose $n$ times a random ball from this urn ...
2
votes
0answers
67 views

Partial sum involving harmonic numbers

QUESTION: I need to know how to compute the partial sum $$\sum_{k=1}^n \frac{H_{k+1}^2-H_{k+1}^{(2)}}{k+2}$$ in terms of the generalized harmonic numbers $H_n^{(m)}$. CONTEXT: This problem arose ...
2
votes
0answers
823 views

Find a sum of a convergent series

Let $x_n$ be a sequence that is given by the following recursive formula: $x_{n+1} = x_n^2 - x_n +1$, where $x_1=a \gt 1$. Find: $$\sum_{n=1}^{\infty} \frac{1}{x_n}$$ Not sure really how to ...
2
votes
0answers
65 views

Finding basis of coefficients satisfying system of linear equations

Setup: Let $C_{i,j}$ denote coefficients which are symmetric in their indices---i.e., $C_{ij}=C_{ji}$. These coefficients are required to satisfy the following relation for all $m\in \mathbb{Z}^+$ ...
2
votes
0answers
56 views

Regularizing the sum of integers with the aid of the Taylor expansion

One way to understand how the sum of all factorials $$ S_{\mathbb{N}!} = 1! + 2! + 3! + \ldots $$ can be regularized comes from an observation that the function $$ f(x) = \text{P.V.}\int \limits_0^\...
2
votes
0answers
50 views

How to compute $\sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}p^{n-2k}(1-p)^{2k}$?

I would like to compute (have a closed form expression) the following sum: $$\sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}p^{n-2k}(1-p)^{2k},$$ $p\in (0,1)$. I know that $$1=(1-p+p)^n = \sum_{k=0}^{n}...
2
votes
0answers
246 views

Sum of all permutations of degree $n$

Find the sum of the permutations of degree $n$. For example, $n = 3$ would give: $123 + 132 + 213 + 231 + 321 + 312 = 1332$ I posed this problem to myself, and was able to solve it (I have put my ...
2
votes
0answers
48 views

Evaluating the expected value of $\log(g(x))$.

I am trying to evaluate the expected value of $\log(x+r)$ with respect to a negative binomial distribution. In other words, I am searching for a closed form or approximation to the quantity \begin{...
2
votes
0answers
51 views

On an infinite summation

I was looking at this problem that can be found at the book "Experimental Mathematics" by Borwein. Evaluate the series $$\mathcal{S}= \sum_{\left ( m, n, p \right ) \in \mathbb{Z} \setminus \{ (0, ...
2
votes
0answers
67 views

Finding the explicit formula for the sum: $\sum_{k=1}^{\frac{n-1}{2}}\sum_{m=1}^{n-1}(-1)^mf_n(kn -\frac{ m (m + 1)}{2})$

I am looking for the explicit formula for the $$\sum_{k=1}^{\frac{n-1}{2}}\sum_{m=1}^{n-1}(-1)^mf_n\Big(kn -\frac{m (m + 1)}{2}\Big)$$ where the $f_n$ is a custom function, with condition $f_n(k)=0, ...
2
votes
0answers
49 views

Understanding the summation symbol

In our script, we have following use of the summation symbol: $(*) \sum_{i=1}^2(a_i+c)$, c is some constant. I know that this is supposed to summarize $a_1+a_2+c$, however the way it is written with ...
2
votes
0answers
52 views

Resource for practicing sums

Frequently I come across a sum (either finite or infinite, and often in the context of moments of discrete random variables) and unless I can force it into one of the few forms that I know, such as a ...
2
votes
0answers
121 views

Is this proof for the “Poisson Summation Formula” correct?

So I was snooping around Proof Wiki for the Poisson Summation Formula proof and came across this: https://proofwiki.org/wiki/Poisson_Summation_Formula In the end it makes the statement: Therefore ...