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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0answers
24 views

Solving for parameter inside summation

What possible methods to solve for parameter inside summation? I arrive at two sums and want find common parameter $a$, which depends on sum index $n$, $a = a(n)$. The sums should hold for all x. ...
1
vote
0answers
26 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
124 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
2answers
115 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 ...
1
vote
0answers
13 views

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

I want to evaluate the limit $\lim_{n \to \infty} \sum_{k=2}^n \frac{n!}{(n-k+1)!} \frac{1}{n^k} c^k$ Any ideas on how to do this?
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2answers
41 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 $\lim_{N\rightarrow\infty}S(N)$ ...
0
votes
0answers
17 views

Algorithm to find solution of 1[U1]+2[U2]+3[U3]+4[U4]+5[U5] + … + k[Uk] = S [on hold]

the algorithm will solve U1,U2,U3... Uk I know the number Uk not null this is a variant of the subset sum problem. Any ideas ? thanks ?
1
vote
2answers
26 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 ...
1
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0answers
35 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
31 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
13 views

Infinite sum of inverse trigonometric function [on hold]

$$\sum_{n=0}^{\infty} atan((acot(n^2+n+3))/(1+acot(n+1)acot(n+2)))$$
4
votes
3answers
76 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
42 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
22 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
28 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
31 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 ...
2
votes
3answers
53 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: $$ \lim_{n \to \infty} \frac{1}{n^{2013}} \sum_{k=1}^n k^{2012} $$ ...
3
votes
1answer
49 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
686 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
16 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
24 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 ...
4
votes
2answers
43 views

prove simple sum, combinatorics

I want to prove that $\sum_{i = 1}^{n} \binom{n}{i}\binom{n}{i-1} = \binom{2n}{n-1}$ On the right hand side we simply have the coefficient of $x^{n-1}$ of the term $(1+x)^{2n}$ But on the other ...
15
votes
1answer
907 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 ...
0
votes
0answers
7 views

Calculate popularity (frequency) of an item sold from multidimensional array [on hold]

I have an array extracted from my database online where i store id status: sold / not sold time: day that the item sold or not price sold and the output example is given below ...
1
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1answer
33 views

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

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 ...
17
votes
5answers
870 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
63 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 ...
1
vote
2answers
82 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
41 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 ...
0
votes
0answers
16 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
2answers
36 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 ...
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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| ...
-1
votes
2answers
24 views

show equivalence of forumulas

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
795 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 ...
4
votes
3answers
94 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
35 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
18 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
0answers
62 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, ...
0
votes
2answers
40 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
votes
1answer
23 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
votes
2answers
19 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
74 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
45 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
21 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 ...