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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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Closed form for $\sum \left (\pm a_1 \pm a_2 \pm \dots \pm a_n\right )^\ell$

I realized that if you take the $2^n$ quantities $$\pm a_1 \pm a_2 \pm \dots \pm a_n$$ and consider the sum of their squares, then the product terms cancel out nicely to give $$\sum \left (\pm a_1 \pm ...
Dumbest person on earth's user avatar
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Applying a summation method to two sums. I need to justify an interchange of summation and integration with this method. Is my use of Fubini flawed?

Below I obtain the Leibniz formula for $\pi$ using a particular summation method. However, you need to justify a step where you interchange a summation and integration. Fubini would not work here. So ...
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Getting the formular of a summation [duplicate]

im kind of stuck at my math homework from my calculus class. To progress further i need to be able to write a Summation into a forumular(?), as seen in the picture. Is there any proven method to do ...
Sicone's user avatar
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Dirichlet's series

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of complex numbers. Furthermore, suppose that exists some $z_0 \in \mathbb{C}$ such that $\sum_{n=1}^{\infty}\frac{a_n}{n^{z_0}}$ converges. Now, my goal ...
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Converting multiple summation into single summation

So I am working on a problem and I am required to convert multiple summations of the following form into a single summation, \begin{align} \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty} \...
Fermion's user avatar
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$A_n-A_{n-1} = q^nA_n-aq^{n-1}A_{n-1}$ for Cauchy sequence

For $|q| < 1$, $|t|<1$, then $$1+\sum_{n=1}^{\infty} \frac{(1-a)(\cdots)(1-aq^{n-1})t^n}{(1-q)(\cdots)(1-q^n)}=\prod_{n=0}^{\infty}\frac{(1-atq^n)}{(1-tq^n)}=\sum_{n=0}^{\infty}A_nt^n$$ Fromwhat ...
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Examine the convergence of $\sum_{n=1}^{\infty}\frac{1}{\sum_{k=1}^{n}k^{1/k}}$ [duplicate]

I know that $$\sum_{n=1}^{\infty}{n^{1/n}}$$ diverges, since the limit when $n$ tends to infinity of $n^\frac{1}{n}$ is equal to $1$. With this, I cannot conclude anything about the convergence of $$\...
Rúben Reis's user avatar
3 votes
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Double Abel summation

I am trying to calculate the Abel summation formula for a function of the form $$ \sum _{j=1}^x \sum _{i=1}^x \phi \left( x-i j \right) $$ where the function $\phi$ meets the requirements for Abel's ...
Richard Burke-Ward's user avatar
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Cesaro summation for matrices

I wonder if Cesaro summation for matrices is the same as summation for sequences. Cesaro summation for sequences means convergence of the arithmetic means (averages) of partial sums of sequence. For ...
Konstantin's user avatar
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Is there an easy way to sum $\sum_{1 \leq i < j < k \leq n} 1$?

I'd like to have a standard procedure to sum terms like $$\sum_{1\,\leq\, i\, <\, j\, <\, k\, \leq\, n} 1$$ without having to "telescope" the sum, beggining from the outermost one and ...
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Regularizing a divergent sum

I have a sum of an infinite series $$ S = \frac{1}{3} - 4 + \frac{196}{15} - 21 + 27 - 33 + 39 - 45 + 51 - 57 + 63 + ... $$ which appears to diverge. This can be separated as such $$ S = (\frac{1}{3} ...
user8675309's user avatar
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How to make sense of rearranging summation symbols in Sheldon Axler's Matrix Multiplication?

I am reading Sheldon Axler's "Linear Algebra Done Right, Third Edition". I am looking at the page where he defines matrix multiplication. He shows the motivation behind the way it is defined,...
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How to evaluate the following sum

While studying sequences and series, I came across summations of geometric series. I am able to derive the sum of a geometric progression and that of arithmetico–geometric sequence. But what do you ...
RajaKrishnappa's user avatar
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Summation with inner products: properties and rearrangement

OPTION 1. I have this expression, $$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l<j}^{L}a_l^2 \frac{a_j}{a_l}$$ and I would like to take out $$\sum \limits_{l=1}^{L}a_l^2$$, as to obtain something ...
CafféSospeso's user avatar
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Generalisation of a multiple summation involving $i<j<k$

I've been looking at a certain type of sum, the first case being $\sum_{i<j} {a_i a_j}$, and trying to simplify it/generalise it. It would be far more useful to write them more explicitly, in terms ...
Habeeb M's user avatar
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Summation of Roots of Cubic Equation

I was attempting the recent May/June 2023, CAIE past paper for Further Mathematics variant 12. The question states: 2     The cubic equation $x^3+4x^2+6x+1=0$ has roots $\alpha$, $\beta$, $\gamma$. ...
BeaconiteGuy's user avatar
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1 answer
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Evaluation of an indexed sum. [closed]

I am going through a textbook solution to an induction problem and I'm not sure why it says this summation holds. $\sum_{m=0}^0 \sum_{k=0}^m a_{m - k, k} = a_{0, 0}$ This is very simple I know, but ...
Newbie1000's user avatar
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Efficient calculation for Lerch Transcendent Expression

I've encountered: $$\Phi(z, s, \alpha) = \sum_{k=0}^\infty \frac { z^k} {(k+\alpha)^s}.$$ When trying to compute: $$\frac{1}{x}\sum_{p=0}^m \frac{2}{(2p-1)\ x^{2p-1}}\ s.t. x\in\mathbb{N} =\ ???$$ ...
user3108815's user avatar
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Is there an identity for $\sum_{k=0}^{n-1}\csc(w+ k \frac{\pi}{n})\csc(x+ k \frac{\pi}{n})\csc(y+ k \frac{\pi}{n})\csc(z+ k \frac{\pi}{n})$?

What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$...
onepound's user avatar
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Follow-up question on Abel Summation

This is a follow-up to a much simpler question I asked here, which @PrincessEev answered promptly and perfectly. She showed me how to rewrite the sum $\sum _{i=1}^x \phi (x-i)$ in such a way that Abel ...
Richard Burke-Ward's user avatar
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Can I write an Abel Summation Formula for this?

Given appropriate constraints, and a continuously differentiable real-valued function $\phi (x)$, the Abel Summation Formula (Wikipedia article here) can be written as $$\sum _{k=1}^x \phi (k) = \...
Richard Burke-Ward's user avatar
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52 views

Fubini theorem in non classical summation methods

I am aware that in the theory of classical infinite sums , one can not generally interchange the order of a double sum or do other infinite sum manipulations. However, these infinite sum manipulations ...
Amr's user avatar
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How do I convert a function to sigma notation when there is a nested and lagging summation within the function?

I am trying to simplify the following function into summation form: $$f(x) = \frac{x_1}{x_0}+\frac{x_2}{x_0-x_1}+\frac{x_3}{x_0-x_1-x_2}+...+\frac{x_n}{x_0-x_1-x_2-...-x_{n-1}}.$$ However, I am unsure ...
brycon2's user avatar
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Is there an identity for $\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$?

What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$$ here it can be shown that where $x=y$, $$n^2 \csc^2(nx) = \sum_{k=0}^...
onepound's user avatar
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Symmetric sum of $ab(3a+c)$

I'm having difficulty with determining the $\sum_{sym}ab(3a+c)$ involving $3$ variables $a,b,c$. I know the cyclic sum is going to be $ab(3a+c)+bc(3b+a)+ac(3c+b)$, but I'm confused about the symmetric ...
math_learner's user avatar
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1 answer
272 views

If $\sum_{p=0}^{2020}{\sin(2^p\theta)\sec(3^{p+1}\theta})=a\tan(b\theta)+c\tan(d\theta)$, find $ac,ad,cd,bd$

If $\sum_{p=0}^{2020}{\sin(2^p\theta)\sec(3^{p+1}\theta})=a\tan(b\theta)+c\tan(d\theta)$, find $ac,ad,cd,bd$ I converted secant to cosine, opened the sigma and tried to take LCM pairwise. But couldn'...
aarbee's user avatar
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2 votes
2 answers
452 views

How can I write the following series in summation notation?

I am writing up a report for my linear algebra class where I am using series to describe certain transformations in a particular vector space: One example is: $$t_2 + t_5 + t_8 + ... + t_{n-1} = \sum_{...
user129393192's user avatar
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Summation with products of values with different powers

For an exercise I am working on I have the following equation, $$ c\sum_{n=1}^{\infty}n\sum_{j=1}^{n}(\frac{\lambda}{2\mu})^{j}(\frac{\lambda}{3\mu})^{n-j}, $$ where $\lambda<\mu$. I have been ...
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Sum of a sum: $\sum_{n=1}^{11}n\left[\frac{1^2}{1+n}+\frac{2^2}{2+n}+\cdots+\frac{11^2}{11+n}\right]$

The given sum is: $$\sum_{n=1}^{11}n\left[\frac{1^2}{1+n}+\frac{2^2}{2+n}+\cdots+\frac{11^2}{11+n}\right]$$ I tried simplyfing it as: $$1^2\left[\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{12}\right] + 2^...
algorhythm's user avatar
1 vote
2 answers
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What computational shortcut finds the sum all possible products given any list of n random real numbers taken r at a time? Here's what I tried...

I have the following computational shortcuts for any list of $n=4$ quantities taken r at a time. My goal is to do this for lists of any length. Taken r at a time, what function can similarly output ...
Tim Lazarus's user avatar
3 votes
1 answer
221 views

Analytical evaluation of infinite series

I am trying to calculate the infinite series $$\sum _{n=-\infty }^{\infty } \frac{(-1)^{n+1} e^{-(n-1)^2\pi}}{1-e^{ (2 n-1)\pi}}\simeq -0.0903244354808$$ Are there any any analytical methods to ...
El Rafu's user avatar
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1 vote
1 answer
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How to express this into a double summation format

Suppose, I am having difficulties writing down as double summation. The expression is $$b_1(t_n)\sum\limits_{k = 1}^1 {a_{1k}u_k}+b_2(t_n)\sum\limits_{k = 1}^2 {a_{2k}u_k}+ ... +b_p(t_n)\sum\limits_{k ...
ash1's user avatar
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1 vote
2 answers
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Summation of $\sum_{r=1}^n \sin(2r-1)\theta$

In a CIE A Level Further Mathematics question paper, the following question appeared: By considering $$\sum_{r=1}^n z^{2r-1}$$ where $z=\cos\theta +i\sin\theta$, show that if $\sin ≠ 0$, $$\sum_{r=1}^...
Musab Usman's user avatar
2 votes
2 answers
248 views

How to prove identity $\sum_{k=0}^{n-1}\tan^2\left(\theta+{k\pi\over n}\right)=n^2\cot^2\left({n\pi\over2}+n\theta\right)+n(n-1)$?

Looking at Jolley, Summation of Series, formula 445: $\sum_{k=0}^{n-1}\tan^2\left(\theta+{k\pi\over n}\right)=n^2\cot^2\left({n\pi\over2}+n\theta\right)+n(n-1)$ How can one prove this? Considering $\...
onepound's user avatar
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Summation in closed form

I have the following summation: \begin{equation} f(k_1,k_2) = \sum_{l_1 = 0}^{k_1}\sum_{l_2 = 0}^{k_2} {\rho}^{l_1+l_2}\delta[k_1-k_2+l_2-l_1] \end{equation} where $\rho$ is a constant and $\delta$ is ...
lord voldemort's user avatar
10 votes
1 answer
304 views

Is there an identity for $\sum_{k=0}^{n-1}\tan^4\left({k\pi\over n}\right)$?

Is there a simple relation for $\sum_{k=0}^{n-1}\tan^4\left({k\pi\over n}\right)$ like there is for $\sum_{k=0}^{n-1}\tan^2\left({k\pi\over n}\right)$? Looking at Jolley, Summation of Series, formula ...
onepound's user avatar
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limit of $\sum_{k=0}^{n} q^{(2k+1)^{3/2}}$ with $0<q<1$?

It is easy to prove that for $0<q<1$, $\sum_{k=0}^{n} q^{(2k+1)^{3/2}}$ converges because it is always increasing and it is always smaller than the convergent series $\sum_{k=0}^{n} q^{2k+1}$ ...
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Is there a formula for $\sum_{i=1}^n a^{1/i}$?

Just what the question says. $\sum_{i=1}^n a^{i}$ has quite well known equation (geometric series). I couldn't find any resources for power $ {1/n} $ though. On a slight tangent; I tried solving it ...
P Shreyas Shetty's user avatar
3 votes
3 answers
136 views

How can I intuitively get from the below summation to a generating function without knowing key identities before hand?

In class, our professor was very adamant that the following simplification is intuitive: \begin{align*} \sum^{\infty}_{n=0}x^n\binom{2n}{n}=\frac{1}{\sqrt{1-4x}} \end{align*} I can get from the RHS ...
Hot Tamale's user avatar
5 votes
1 answer
116 views

Showing $\lim_{x\rightarrow 1^-} \sum_{k=0}^\infty (-x)^k (2k+1) \ln(2k+1) = \frac{-2C}{\pi}$

While I was playing around with divergent summation, I noticed that the following appears to be true: $$\lim_{x\rightarrow 1^-} \sum_{k=0}^\infty (-x)^k (2k+1) \ln(2k+1) = \frac{-2C}{\pi}$$ where $C = ...
user196574's user avatar
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2 votes
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Borel summation of a divergent series as the best estimate of a function

My current research has me working with series of the form $\sum_{n=0}^\infty f_n z^n$, $z\in\mathbb{C}$ that usually have zero convergence radius (they diverge for all values of $z$). Due to ...
Néstor González Gracia's user avatar
1 vote
1 answer
76 views

Summation Notation for Current Index

I have a question on how to properly present a summation with the use of variables. I am not so familiar with how to properly use summation notation, and am looking for some advice on how to do so ...
Justin's user avatar
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2 votes
1 answer
193 views

Watson-Nevanlinna theorem for $e^{-1/z}$

I am currently trying to understand Watson-Nevanlinna (WN) theorem, which gives sufficent conditions for a function $f(z)$ to be equal to the Borel sum of its asymptotic expansion as $z\to0$. The ...
Néstor González Gracia's user avatar
1 vote
2 answers
72 views

Finding the sum of finite geometric series

I'm doing the following summation $\sum_{l=k}^{n}2^l$ $\sum_{l=k}^{n}2^l = 2^k + 2^{k+1} + 2^{k+2} + \ldots+ 2^{n-1} + 2^{n}$ $S_n=a_1\dfrac{1-r^n}{1-r} \therefore S_n=2^k\dfrac{1-(2)^n}{1-2} = 2^{k+...
Carlos's user avatar
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1 vote
1 answer
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How to factorize and solve equations with $\Sigma$ notation?

I have a few doubts about the properties of sigma notation, $\Sigma$ . My questions rely on factorization and solving equations with $\Sigma$.On account of the fact that my questions are correlated, I ...
user avatar
0 votes
2 answers
292 views

How do I solve the double summation $ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^2 - n^2}{(m^2 + n^2)^2}$?

Basically I'm stuck with this double summation. I want some help evaluating this summation. $$ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^2 - n^2}{(m^2 + n^2)^2} $$ Am I allowed to change the ...
Vaibhav C M's user avatar
5 votes
3 answers
141 views

How does the divergent sum $\sum_{n=1}^\infty\cos(2n\gamma)\sin(2nt)$ correctly evaluate an integral? Surely distributions don’t apply here

$\newcommand{\d}{\,\mathrm{d}}\newcommand{\res}{\operatorname{Res}}$Note: I don’t know any distribution theory myself, but I was informed by someone else and hinted to by this answer that my problem ...
FShrike's user avatar
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2 votes
0 answers
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Closed form of $\sum _{k=1}^{n }\:\lfloor{n/k}\rfloor$ [duplicate]

I came across a question in which, if I am able to calculate this sum, $\sum_{k=1}^{n }\:\lfloor{n/k}\rfloor=$ ? it would get solved quite easily. I have never seen any closed form for this question,...
shsh23's user avatar
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1 answer
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Proving $\sum_{k=1}^{n}\frac{1-a\cos\left(\frac{2\pi k}{n}\right)}{1-2a\cos\left(\frac{2\pi k}{n}\right)+a^{2}}=\frac{n}{1-a^{n}}$

How do prove this relation? $$\sum_{k=1}^{n}\frac{1-a\cos\left(\frac{2\pi k}{n}\right)}{1-2a\cos\left(\frac{2\pi k}{n}\right)+a^{2}}=\frac{n}{1-a^{n}} \text{ } \text{ } \text{ } \text{ } \text{ }\...
Asmat Qatea's user avatar
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Notions of convergence of real sequences. Where can I learn about this?

I am looking some good references to introduce myself to convergence process of real sequences. I know classical convergence, uniform convergence, almost convergence, matrix summation method, ...
Senna's user avatar
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