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

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3answers
36 views

Can I approximate a series as an integral to find its limit and determine convergence?

Find $\lim \limits_{n \to \infty} (a_n)$, where $a_n=\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+...+\frac{n}{n^2}$. So I can solve it like that ...
0
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1answer
92 views

Does this infinite sum converge?

Does this infinite sum converge? I have tried many methods that I know. $$\sum_{n=1}^\infty\frac{\left(1-\frac1n\right)^{n^2}}{3n^2+2}e^n$$ edit : after I saw the comments, I tried the ratio method ...
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1answer
13 views

Is $\sum_{k=0}^{\infty}\frac{(-t)^k B^{k-l-1}}{k!}$ summable in a closed form

I am wondering if $$\sum_{k=0}^{\infty} \frac{(-t)^k B^{k-l-1}}{k!}$$ is sumable in a closed form. We have $t\in \mathbb R$, $l\in \mathbb N$. For $B$ I am interested in two cases: $B\in \mathbb R$ ...
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0answers
25 views

What's about $ \sum_{n=1}^{\infty} \frac{ \mu\left( \sigma (n)\right)}{n^3} ,$ where $\mu(n)$ is Möbius function and $\sigma(n)=\sum_{d\mid n}d$?

Let $ \mu (n)$ the Möbius function and $ \sigma (n)$ the sum of divisors function, then the arithmetical function $g(n)= \frac{ \mu\left( \sigma (n)\right)}{n^3} $ isn't multiplicative since ...
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3answers
87 views

Show that $\sum\limits_{k=1}^{\infty}\ln\left(\frac{k(k+2)}{(k+1)^2}\right) = -\ln(2)$?

I'm working with the following sum and trying to determine what it converges to: $$\sum_{k=1}^{\infty}\ln\left(\frac{k(k+2)}{(k+1)^2}\right)$$ Numerically I see that it seems to be converging to ...
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1answer
38 views

Evaluate the limit of the following summation.

I came across this question and i am stuck at this. Evaluate $\sum_{k=1}^n \frac{k}{k^4+k^2+1}$ where $n$ goes to infinity. any ideas?
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0answers
17 views

How is Abel's Summation Formula being applied here?

I am confused as to how the second equality goes through? For those who are interested, this is from the http://quant-econ.net/py/amss.html.
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2answers
33 views

Proof of the binomial identity $\displaystyle\binom{m}{n}=\sum_{k=0}^{\lfloor n/2 \rfloor} 2^{1-\delta_{k,n-k}} \binom{m/2}{k} \binom{m/2}{n-k}$

Trying to prove some uncorrelated things, I came across the following identity: $$\binom{m}{n}=\sum_{k=0}^{\lfloor n/2 \rfloor} 2^{1-\delta_{k,n-k}} \binom{m/2}{k} \binom{m/2}{n-k}, $$ where ...
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4answers
53 views

Find $k$ in arithmetic progression knowing $a_4$, $n=10$ and knowing fact of $S_{k\,\text{last}}$

I know that an arithmetic serie has $10$ terms and some more things: $$a_4=0$$ $$\;\quad\qquad\qquad\qquad n= 10 \quad\text{As I said above}$$ $$S_{k\,\text{last}} = 5S_{k\,\text{first}}$$ In other ...
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2answers
272 views

Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let ...
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0answers
37 views

$4$ or more type $2$ implies $3$ or less type $1$

I'm having difficulties with the logic with the last part of the reformulation part of the problem below. Let $x_i$ be the the number of ships of type $i$ to purchase. For $4a:$ (the ...
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0answers
11 views

Summation involving Hermite polynomials.

I am wondering how to calculate the following summation $$\sum_{j=0}^{\infty}\frac{e^j}{j!\sqrt{2^j}}H_j(x),$$ where $H_j(x)$ are the Hermite polynomials. I thought that we could use the generating ...
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6answers
215 views

Proof there is no way to chose signs to make sequential sum $1+2+3+\cdots+10$ even [closed]

I've figured that for the sum $$1+2+3+4+5+6+7+8+9+10=55$$ There is no way to chose the signs of the numbers to get an even sum. I'm really struggling to prove this and would appreciate some ...
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1answer
38 views

Asymptotics of a series involving cos integral functions

I'm looking for the asymptotic expansion( or value ) of the following function \begin{equation} F[y,t] = \sideset{}{'}\sum_{n \in \mathbb{Z}}\text{Ci}\big[\frac{n^2}{t}\big] - ...
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1answer
19 views

(Resolved) Does the sum of a subset of the Harmonic sequence converge iff its density approaches 0?

Update: This question has been resolved. I have made some mistakes in this post. I will leave my post here for readers to find out my mistakes. I have noticed that the post is a bit too long. So if ...
1
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0answers
7 views

Transforming a polynomial sum using a series expansion (BCH codes)

In my study of BCH codes I've come across the following equation (the "key equation"): $$ \Omega(x) \equiv \Lambda(x)S(x) \mod x^{n-k} \tag{1} $$ Where the two terms on the right are defined by: $$ ...
12
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3answers
141 views

Prove $\sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}.$

How to prove $\displaystyle \sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}$ for $1 \leq \alpha \leq p$? EDIT: This is a result that I derived ...
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3answers
713 views

Interesting representation of $e^x$

So I discovered the following formula by using the Taylor series for $\ln (x+1)$ $$x= \ln ...
1
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1answer
23 views

Recursion solution doesn't seem correct

I'm studying real analysis at the moment on my own. So I don't have a professor to ask if I'm not sure about my answer to an exercise from my text. So I'll ask you guys. The question is Let $d$ ...
2
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0answers
39 views

Representation of e

I was on this wikipedia page https://en.wikipedia.org/wiki/List_of_representations_of_e which has a list of representations of the constant e. I came across this one representation that looked ...
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0answers
14 views

Weighted logarithmic ranking

I want to have a ranking of players by percentage of shots made, weighted by the total number of shots attempted. The weighting should follow a log scale, so for example Player A has 100% accuracy, ...
2
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2answers
31 views

Gamma representation of certain sequence

I'm trying to find a gamma rep for $ 15 \cdot 13 \cdot 11 \cdot 9 \cdot 7 \cdot ... $ Steps so far: It's a simple sequence of $ n \cdot (n-2) \cdot (n-4) \cdot (n-6) \cdot (n-8)... $ and so on. ...
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2answers
30 views

Prove by induction: the coefficients of (a+b) to the power of n are the same if turned into a number as 11 to the power of n

Proof by induction that the coefficients of $(a+b)^n$ in order, if place as a number, the first coefficient being having the biggest place value, and each number lowers in place value, are equal to ...
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3answers
60 views

Prove $\sum_{k=0}^{n}{n \choose k}(-1)^k \frac{1}{k+1} = \frac{1}{n+1} $

I am going through a proof of the combinatorial result $$\sum_{k=0}^{n}{n \choose k}(-1)^k \frac{1}{k+1} = \frac{1}{n+1} $$, and have found something I don't understand. The proof is as follows: ...
2
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1answer
28 views

Cascading Summation (2) $\sum_{i_1\le i_2\le i_3\le \cdots \le i_m}^n \left[\prod_{r=1}^m (i_r+2r-2)\right]/(2m-1)!!$

Evaluate the following summation: $$\large\sum_{i_1\le i_2\le i_3\le \cdots \le i_m}^n \left[\prod_{r=1}^m (i_r+2r-2)\right]\bigg /(2m-1)!!\\ ...
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0answers
14 views

Rearranging summation terms including a complex exponential expression

I'm reading a paper on signal processing and having a hard time wrapping my head around a step the author takes. The signal of interest is defined as $r_k = e^{j(2\pi\Delta f k T_s + \theta)} + v_k$ ...
6
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1answer
58 views

Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$ [duplicate]

Combinatorial argument for $$1+\sum\limits_{r=1}^{r=n} \ r\cdot r! = (n+1)!$$ The algebraic proof is easy as $r=(r+1)-1$.
6
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1answer
44 views

Sum of the reciprocal of the prime-position primes.

The primes are $2, 3, 5, 7, 11, 13...$ The sum of the reciprocals of the primes diverges, proven by Euler: $$\sum_{n=1}^\infty{\frac{1}{p_n}}=\infty$$ Here, $p_n$ is the $n$-th prime. I'm asked to ...
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0answers
20 views

Sum notation $\sum_{\sigma\in\{\pm 1\}^n}$?

I would like to know what the following sum notation means: $$\sum_{\sigma\in\{\pm 1\}^n}\left(\prod_{1\leq i\leq n}F(x_i^{\sigma_i})\right)$$ where $n$ is a positive integer, $x_i$ are some ...
0
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1answer
31 views

Can you take the limit as $x \to \infty$ of an expression such as $\sum_{n \in \mathbb{Z}} \ln(|x - n|)$? [closed]

Consider the function $f(x) = \sum_{n \in \mathbb{Z}} \ln(|x - n|)$ I'm not really concerned with its convergence properties, what I am concerned with is if it is possible to take the limit as $x ...
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2answers
51 views

Evaluate the following sums using generating functions

I have two series that I'm supposed to evaluate using generating functions. (a) $0+1+2+3+4+ ...+ n$ (b) $0 + 3 + 12+...+3n^2$ I know how to evaluate (a) using walks in Pascal's triangle: the answer ...
2
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2answers
92 views

Cascading Summation $\sum_{i=1}^n\sum_{j=i}^n\sum_{k=j}^n \frac {i(j+2)(k+4)}{15} $

Evaluate $$\sum_{i=1}^n\sum_{j=i}^n\sum_{k=j}^n \frac {i(j+2)(k+4)}{15} $$ Background Many basic summation questions on MSE relate to a single index - it might be interesting to devise a ...
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0answers
14 views

How do you expand two sigma(summation) signs?

Consider the figure below. Do I consider the expression with the second sigma sign as being put inside the first sigma sign along with other terms or should I consider it the result of two separate ...
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1answer
25 views

Infinite sum over Gamma functions?

I am having quite a bit of trouble understanding this sum. Can someone explain to me exactly how to this from 1 to 3,very easily way? Question its from this webpage Thanks.
4
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2answers
119 views

Infinite Sum without using $\sin\pi$

What's a purely algebraic way to prove that $\pi-\frac{\pi^3}{3!}+\frac{\pi^5}{5!}-\dots=0$? I'm sure that the first step is to write $\pi=4-\frac43+\frac45-\dots$, but I haven't been bold enough to ...
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2answers
22 views

Double Sigma with nested index, $i$

I am trying to solve this double sigma but my answer doesn't seem right. $$\sum\limits_{i=1}^n \sum\limits_{j=i}^{n^2}1=-i\sum\limits_{i=1}^n n^2 = -in^3$$
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1answer
80 views

Trigonometric proof stuck with induction step

I am trying to prove: $$\sum_{s=0}^{\infty}\frac{1}{(sn)!}=\frac{1}{n}\sum_{r=0}^{n-1}\exp\left(\cos\left(\frac{2r\pi}{n}\right)\right)\cos\left(\sin\left(\frac{2r\pi}{n}\right)\right)$$ We know that ...
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1answer
20 views

Number of lattice points in triangle formed by x-axis, y-axis and given line

Given a line $ax+by=c$ where $a,b,c$ are positive integers. Is there any formula to find the number of points inside the triangle formed by this line, $x$-axis and $y$-axis? Points on the boundary ...
0
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1answer
19 views

Turning points of a weighted cosine basis sum

I'm doing some work with a cosine basis, where in the interval $[0, \pi]$ some function $f(x)$ is given by $$ f(x) = \sum_{n = 0}^{M} a_n \cos{\left( nx \right)} $$ For a given set of coefficients ...
2
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0answers
37 views

Pull constant out of a summation of fractions

General problem $$ \sum_{i=1}^n \frac{a_i + x}{b_i + x} = 0 $$ Is it possible for solve for $x$? Some context I've hit a road block in my derivation... At this point, I need to pull the model ...
1
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2answers
38 views

Changing summation in a power series

I'm doing a question in my power series unit that involves adding summations together, I just started this unit so I'm not totally clear on how changing summation works, from what I understand you ...
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3answers
53 views

Find explicit formula for summation

I have this summation: $\displaystyle\sum_{i=1}^{\log_2 n} 2^{i}$, any suggestion of how get an explicit formula?
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1answer
106 views

Summation notational convention

Please correct improper notation/terminology $$\sum_{k=0}^{n-1} ar^k$$. $$\sum_{k=1}^{n} ar^{k-1}$$ As far as I can tell these both represent the same thing. It's the partial sum {$S_n$} where the ...
2
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1answer
21 views

Solving differential equations problem with Unit-step Function and Delta Function.

Suppose we have a monic differential equation defined by $$x'' + x = \sum_{n=0}^\infty \delta(t - 2n\pi)$$ When we take the Laplace transform of this differential equation, we get $$s^2X(s) + X(s) = ...
2
votes
3answers
80 views

Sum of a series $\frac {1}{n^2 - m^2}$ m and n odd, $m \ne n$

I was working on a physics problem, where I encountered the following summation problem: $$ \sum_{m = 1}^\infty \frac{1}{n^2 - m^2}$$ where m doesn't equal n, and both are odd. n is a fixed constant ...
0
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0answers
30 views

How to induct for double summation?

I have no idea on how to approach this? $$ \sum_{i=1}^{n}\sum_{j=1}^{m}a_i + a_j = \sum_{i=1}^{n} a_i + \sum_{i=1}^{m} a_j (it \space may \space be \space wrong \space it \space is \space just ...
4
votes
3answers
120 views

Upper bound for $\sum_{n=1}^xn^{k-1}$

From some Calculus and guess-work, I found that $$k\sum_{n=1}^xn^{k-1}<(x+\frac12)^k\tag1$$ In fact, I found that it was very, very, close. And, from even more guesswork, ...
0
votes
1answer
30 views

Nice formula for a sum product

So suppose I have an ordered set of numbers: $(a_1, a_2, ..., a_n)$ and I want to express the following sum/product in an elegant manner: $ a_1 + a_1 a_2 + a_1 a_2 a_3 + ... + a_1 a_2 ... a_n $ I ...
0
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0answers
25 views

Resolving Zeros in Product of items in list.

Given the formula: $\sqrt [ 1/N ]{ \prod _{ n=1 }^{ N }{ { P }_{ n } } } $ where ${ P }_{ n }$ is a list of real numbers, e.g. [0.4, 0.3, 0.2, 0.1] And the ...
4
votes
3answers
185 views

Find formula for $\frac{1}{\sqrt 1}+ \frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n}$

I have the series: $$\frac{1}{\sqrt 1}+ \frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n}$$ I find hard to generalize into one formula, any explanation would be helpful.