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Questions tagged [summation-method]

Use for methods for constructing generalized sums of series, generalized limits of sequences, and values of improper integrals.

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33 views

Sum of reciprocal sine function $\sum\limits_{k=1}^{n-1} \frac{1}{\sin(\frac{k\pi}{n})}=?$

The question comes to me when I find there are answers on summation of some forms of trigonometric functions, i.e. $$ \sum\limits_{k=1}^{n-1} \frac{1}{\sin^2(\frac{k\pi}{n})}\\ \sum\limits_{k=0}^{n-1}...
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1answer
33 views

Summation of a sequence when sum of previous terms is inside sigma

In the following summation, I need the current sum to be a part of the computation in the actual sigma, represented as, $$a_n = \sum_{i=1}^n {[a_{n-1}(i+1) + i]}$$ for example, $$a_1 = \...
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1answer
50 views

Find $\sum_{i=10}^n (\sum_{r=10}^i \binom{30}{r}\binom{20}{r-10})$=?

Evaluate: $$\sum_{i=10}^n \left(\sum_{r=10}^i \binom{30}{r}\binom{20}{r-10}\right)$$ I tried this and my result come: $$n\binom{30}{10}\binom{20}{1}+ (n-1)\binom{30}{11}\binom{20}{2} + (n-2)\binom{30}...
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2answers
41 views

Why is the borel sum analytic

I am currently reading in a book about Borel sums as a method of analytic continuation of power series. So given a power series $\sum_{n=0}^{\infty}a_nz^n$ the borel sum is defined as $\int_0^\infty e^...
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1answer
24 views

Find the summation of given expression

I am trying to solve the following question which is Ex3 from Arthur Engel, Problem Solving strategies. Here is the question: $\sum_{k=1}^n k^3 {n \choose k}$ and asks to find the sum. I am sincerely ...
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2answers
40 views

How do I notate this?

Let’s say that I have a whole pie, and I take $75\%$ (or 3/4ths) of that pie. Then, I take $75\%$ of the remaining quarter of the pie and add it to the original $75\%$, I would have $93.7\%$ of the ...
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4answers
49 views

Sum of arithmetic progressions

There is this sum: $$\sum_{i=0}^{n-1}\left(\sum_{j=i+1}^{n-1}(n-j-1)\right)=\frac{1}{6}(n-2)(n-1)n$$ I don't understand how the formula is derived. What I do currently understand is this: For each i,...
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1answer
29 views

inverting product and sum

I am always having a hard time when I am dealing with expressions of the form : $$B(x) = \prod_{i = 1}^N \sum_{k = 1}^{L_i} a_{i, k}(x)$$ where $(L_n)$ is a sequence of naturals numbers such ...
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1answer
46 views

Simplifying a summation ratio

I need some help to simplify the following: $$\frac{\sum_{k=0}^na_nr^n}{\sum_{k=0}^na_n(1+r)^n},$$ where $n \in \mathbb{N}$, $a_n$>0 and $r$>0. More specifically, the question I have is $$\frac{\sum ...
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39 views

How to find summation formula for the following power series?

How to find $\text{summation formula}$ for the following power series? $$ \sum_{n=0}^{\infty} \frac{((an+b)!)^{an+b}}{r+((an+b)!)^{an+b}} p_k(n) \cdot \frac{x^n}{(an+b)!} , \ r \in \mathbb{Q}^{+}, ...
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25 views

Summation with exponential part

Can anyone help me with calculating below summation? $\sum_{n=1}^{\infty} \frac{n^2 e^{-a(n^2+b)}}{n^2+b}$, in which $b$ and $a$ are positive.
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1answer
47 views

Estimate for multiple harmonic sum

I am interested in estimating the following family of sums: $$S_k(n) \equiv \sum_{\substack{n_1, \ldots, n_k \geq 1\\n_1 + \ldots + n_k = n}}\frac{1}{n_1\ldots n_k}$$ where $k \geq 1, n \geq 1$. A ...
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45 views

A sum of series problem with alternating sign of terms

I came across a problem that requires me to find the sum of a series. The term of the series $T_n$ is given by $$T_n = (-1)^{\frac{n(n+1)}2}n^2$$ Sum till $4n$ terms is to be found. Writing down the ...
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1answer
52 views

Sum of an infinite series of fractions involving multiple terms in the denominator [duplicate]

This is the series in question: $$S = \frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4} \ldots$$ The general term seems to be: $$T_n=\frac{n}{1+n^2+n^4}$$ In the original question, which ...
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1answer
30 views

Evaluating Summation of Infinite Series

Let $a_{1},a_{2},...,a_{n}$ be a monotone increasing sequence of numbers satisfying $\sin(a_{k}) = \frac{k}{n}$ and $a_{k} \leq \frac{\pi}{2}$ for all $1\leq k \leq n$. I would like to calculate $\lim\...
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10 views

Stability of matrix summability methods

I'm currently studying summabiltiy method defined by infinite matrices and I'd like to find a characterization of stable methods. Suppose $A$ is an infinite matrix, $x$ a real (or complex) sequence, ...
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1answer
55 views

Is there a Mistake in this article of Barret et al.

In their article "On the spectral radius of a {0,1) Matrix Related to Mertens' Function", Barret et al. assert an inequality just at the end of p. 156. Apparently, this inequality comes from the Abel ...
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15 views

Stochastic Processes problem

I have already solved the problem but only up to the 2nd part, I can't equate directly the 3rd part but i don't know how to explain the reason behind it where k=0 then the greater than or equal to ...
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23 views

Help finding the result of a sum over a lattice with two variables

I would need some help for finding a closed form for $$f(a,b):=\sum_{(k,l)\in\mathbb{Z}^2,\ l\ne0}\frac{1}{(2 i k a \pi + l b) (2 i (k - l) a \pi + l b)}$$ In general, I do not know many ...
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2answers
232 views

Multiplying summation with same indices and limits

What would be $(1-\sum \limits_{k=0}^m x^k )(1-\sum \limits_{k=0}^m y^k ) ?$ I dont understand how can I multiply summation of same indices. I checked "multiplication of finite sum (inner product ...
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126 views

Find the sum of $1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$

Find the sum of $$1-\frac17+\frac19-\frac1{15}+\frac1{17}-\frac1{23}+\frac1{25}-\dots$$ a) $\dfrac{\pi}8(\sqrt2-1)$ b) $\dfrac{\pi}4(\sqrt2-1)$ c) $\dfrac{\pi}8(\sqrt2+1)$ d) $\dfrac{\...
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14 views

Regarding methods of creating more interesting patterns of plus and minus in a summation.

In a summation notation, we can alternate the sign of each term by multiplying the entire sum (or equally, each term) by negative one to the n, n being the term count. My question is if there are ...
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1answer
44 views

Sum to $n$ terms the series $\frac{1}{3\cdot9\cdot11}+\frac{1}{5\cdot11\cdot13}+\frac{1}{7\cdot13\cdot15}+\cdots$.

Q:Sum to n terms the series : $$\frac{1}{3\cdot9\cdot11}+\frac{1}{5\cdot11\cdot13}+\frac{1}{7\cdot13\cdot15}+\cdots$$ This was asked under the heading using method of difference and ans given was $...
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3answers
209 views

Sum the first $n$ terms of the series $1 \cdot 3 \cdot 2^2 + 2 \cdot 4 \cdot 3^2 + 3 \cdot 5 \cdot 4^2 + \cdots$

The question Sum the first $n$ terms of the series: $$ 1 \cdot 3 \cdot 2^2 + 2 \cdot 4 \cdot 3^2 + 3 \cdot 5 \cdot 4^2 + \cdots. $$ This was asked under the heading using method of difference and ...
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1answer
57 views

Sum of the series $(i)$ and $(ii)$

Find the sum of the series: $(i)1+\frac{2}{9}+\frac{2.5}{9.18}+\frac{2.5.8}{9.18.27}+\cdots +\infty$$(ii)1+\frac{3}{4}+\frac{7}{16}+\frac{13}{64}+\cdots +\infty$The answer providing my book is :$(i)\...
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1answer
32 views

Resolving dependencies of nested dependent summations

If I have two or more nested summations in which the inner ones depend on the outer ones, how could I “remove” or resolve the dependencies? In this case, for example: $$ \sum_{i=1}^{n-1} \sum_{j=i+1}...
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4answers
114 views

Getting a closed form from $\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1$

I need to get a closed form from $$ \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1 $$ Starting from the most outer summation, I got $$ \sum_{k=1}^{j} 1 = j $$ But now I don't know how to ...
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1answer
41 views

Finding a closed formula for $\sum_{i=0}^n 2^i \cdot (n-i)$ through the perturbation method

I need to find a closed form for $$\sum_{i=0}^n 2^i \cdot (n-i)$$ Through the perturbatino method. How could I start? May I reduce the summation in multiple simpler summations?
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1answer
55 views

How to show that the following relation is valid for nested summation?

I have seen following relation in a research paper $$\sum_{x_1=1,x_1\neq1}^{K}~\sum_{x_2=x_1+1,x_2\neq 1}^K\cdots \sum_{x_n=x_{n-1}+1,x_n\neq1}^Kf(x_1,x_2,\cdots,x_n)+\sum_{x_1=1,x_1\neq2}^{K}~\sum_{...
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1answer
82 views

How to expand summations in this variance proof

I'm just going through my notes, and noticed I couldn't follow how to do this step, in deriving a formula for the variance. $$ V(Y) = E\left[\sum_i^n a_i (Y_i - \mu_i)\right]^2$$ $$ =E\left[\sum_i^n ...
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3answers
71 views

What's the easiest, most concise, way to prove this simple swapping of order of nested summation?

$$ \sum\limits_{n=1}^{N} \, \sum\limits_{k=1}^{n} \, a_{nk} \ = \ \sum\limits_{k=1}^{N} \, \sum\limits_{n=1}^{k} \, a_{nk} $$ for $N \in \mathbb{N} < \infty$
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0answers
30 views

Crucial step of converting Fourier series to Fourier transform

when I was deriving fourier transform from fourier series, I encoutered a difficulty of thinking in converting the summation into integral (I'm from engineering background). So could someone give some ...
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2answers
2k views

Sum to n terms of the following series: $2 \cdot 2 + 6 \cdot 4 + 12 \cdot 8 + 20 \cdot 16 + \cdots $

I able to get the general term or $\large n$th term of the series is: $$U_n=(n^2+n)2^n$$ Now i want to get $S_n$ by the method of difference.That's why I need to make $$U_n=V_r-V_{r-1}$$ My book is ...
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2answers
225 views

Sum to n terms the series $\cos \theta+ 2\cos 2\theta+ \cdots + n\cos n\theta$

Sum to $n$ terms and also to infinity of the following series:$$\cos \theta+ 2\cos 2\theta+ \cdots + n\cos n\theta$$the solution provided by the book is $$S_n=\frac{(n+1)\cos n\theta-n\cos(n+1)\theta-...
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2answers
201 views

Sum of the series $\frac{1}{2.4.6}+\frac{2}{3.5.7}+\frac{3}{4.6.8}+…+\frac{n}{(n+1).(n+3).(n+5)}$.

Sum to n terms and also to infinity of the following series: $$\frac{1}{2.4.6}+\frac{2}{3.5.7}+\frac{3}{4.6.8}+.....+\frac{n}{(n+1).(n+3).(n+5)}$$ the solution provided by the book is $$S_n=\frac{...
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2answers
110 views

Summation of Double Exponential Series [closed]

Is there any known closed form or tight bound analysis (big-O or big-$\Theta$) for $\sum_{i = 0}^{n} 2^{2^i}$?
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4answers
212 views

Summation method for $1-3+9-27 + \dots $ [closed]

Is there a summation method to compute the series $$1-3+9-27+\dots ? $$ We knew this is divergent in the usual sense, but there are summation methods for giving a sense to divergent series. For ...
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1answer
122 views

How to solve $\sum_{i=1}^{n} \sin(x_i - \mu) = 0$ for $\mu$? (Maximum likelihood estimation)

The distribution is defined as: $f(x; k, \mu) = exp(k\cos(x-\mu))$ (I have removed the term before it because it will be $0$ when we take the derivative of the log). We must find the MLE of $\mu$ My ...
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106 views

Hausdorff methods of summation

From the book of Boss "Classical and modern methods in summability": "The class of Hausdorff methods includes the Hölder, Cesaro and Euler methods. A large number of other matrix methods which play ...
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1answer
95 views

Zeta regularization vs Dirichlet series

Suppose you have a sequence of real numbers, denoted $a_n$. Then the sum of the sequence is $\sum_n a_n$ If this is divergent, we can use zeta regularization to get a sum. We can do this by defining ...
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49 views

Could be this:( $f$ is Lebesgue integral $\iff$ $f$ is Borel-summable ) works?

Borel summation is a summation method for divergent series, It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. and The ...
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204 views

Exercise 2 from Terry Tao's blog on Euler-Maclaurin, Bernouilli numbers, and the zeta function

In the blog post The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, Terry Tao looks at the commonly-cranked 'absurd' formulae $$\begin{align} \...
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155 views

Does Ramanujan summation extend Abel summation?

I was under the impression that if a series is Abel summable, then it is Ramanujan summable to the same value, but when I answered this question and was asked this as a follow-up, I was unable to ...
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123 views

Find the value of $\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots + \frac{1}{1+2+3 +\ldots+2015}$

The question: Find the value of $$\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots + \frac{1}{1+2+3 +\ldots +2015}$$ If this is a duplicate, then sorry - but I haven't been able to find this ...
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2answers
98 views

Closed-form summation of $\sum_{i=1}^n i\frac{x^i}{i!}$

Is there any way to find the closed-form of this finite summation, knowing that x<1? It is part of a larger equation that I am trying to solve/simplify, which has proven to use a lot of theory that ...
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0answers
43 views

Alternative way to write triple summation

I have expression as $$\sum_{l\in \phi}^{} \sum_{i=1}^{K} \sum_{m\in \phi \setminus l}^{} a(l,i) b(m,i) c(l,m,i)$$ where $m\in \phi \setminus l$ means excluding $l$. I would like to write this ...
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1answer
245 views

Summation of series with factorial

enter image description here I tried breaking the terms into differences or finding a generalised term but did not get it right. Can someone please help me to proceed with this?
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1answer
199 views

Compute the following expression $\Sigma {100}{k=1} [k*(-1)^k]$.

I honestly have no idea how to format this. Anyways, I did this so far using the properties listed in my notes. I am pretty sure it is wrong, and if it isn't, where would I go from here?
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80 views

An oscillating series of real numbers which can be resummed to a complex value

Okay so here I go again studying summability theory I was wondering the following problem but first I'll state a few conventions: A series diverges if the partial sums tends to $\pm \infty$, ...
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1answer
377 views

A Summability methods which sum the harmonic series

Studying summability theory I've come across many summation methods however by now I know only two not very interesting method which re-sums the harmonic series $\sum_{n=0}^\infty \frac 1{n+1}$ : the ...