# 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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### 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 ...
1 vote
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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 ...
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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 $\alpha$, $\beta$, $\gamma$. ...
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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 ...
1 vote
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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: $$\frac{1}{x}\sum_{p=0}^m \frac{2}{(2p-1)\ x^{2p-1}}\ s.t. x\in\mathbb{N} =\ ???$$ ...
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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 $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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### 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'...
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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 ...
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### 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 ...
1 vote
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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 $\sum_{k=0}^{n} q^{2k+1}$ ... 29 views

I am doing a problem that needs me to solve the Laplace's equation in a cubric boundary condition using separation of variables. During the procedure solving it, I meet the following series: \... 0 votes 0 answers 42 views ### 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 ... 3 votes 3 answers 131 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 ... 5 votes 1 answer 102 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}$$where C = ... 2 votes 0 answers 83 views ### Borel summation of a divergent series as the best estimate of a function My current research has me working with series of the form \sum_{n=0}^\infty f_n z^n, z\in\mathbb{C} that usually have zero convergence radius (they diverge for all values of z). Due to ... 1 vote 1 answer 55 views ### Summation Notation for Current Index I have a question on how to properly present a summation with the use of variables. I am not so familiar with how to properly use summation notation, and am looking for some advice on how to do so ... 2 votes 1 answer 138 views ### Watson-Nevanlinna theorem for e^{-1/z} I am currently trying to understand Watson-Nevanlinna (WN) theorem, which gives sufficent conditions for a function f(z) to be equal to the Borel sum of its asymptotic expansion as z\to0. The ... 1 vote 2 answers 67 views ### Finding the sum of finite geometric series I'm doing the following summation \sum_{l=k}^{n}2^l \sum_{l=k}^{n}2^l = 2^k + 2^{k+1} + 2^{k+2} + \ldots+ 2^{n-1} + 2^{n} S_n=a_1\dfrac{1-r^n}{1-r} \therefore S_n=2^k\dfrac{1-(2)^n}{1-2} = 2^{k+... 1 vote 1 answer 92 views ### How to factorize and solve equations with \Sigma notation? I have a few doubts about the properties of sigma notation, \Sigma . My questions rely on factorization and solving equations with \Sigma.On account of the fact that my questions are correlated, I ... 0 votes 2 answers 245 views ### How do I solve the double summation  \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^2 - n^2}{(m^2 + n^2)^2}? Basically I'm stuck with this double summation. I want some help evaluating this summation.$$ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^2 - n^2}{(m^2 + n^2)^2} $$Am I allowed to change the ... 5 votes 3 answers 133 views ### How does the divergent sum \sum_{n=1}^\infty\cos(2n\gamma)\sin(2nt) correctly evaluate an integral? Surely distributions don’t apply here \newcommand{\d}{\,\mathrm{d}}\newcommand{\res}{\operatorname{Res}}Note: I don’t know any distribution theory myself, but I was informed by someone else and hinted to by this answer that my problem ... 2 votes 0 answers 72 views ### Closed form of \sum _{k=1}^{n }\:\lfloor{n/k}\rfloor [duplicate] I came across a question in which, if I am able to calculate this sum, \sum_{k=1}^{n }\:\lfloor{n/k}\rfloor= ? it would get solved quite easily. I have never seen any closed form for this question,... 0 votes 1 answer 92 views ### Proving \sum_{k=1}^{n}\frac{1-a\cos\left(\frac{2\pi k}{n}\right)}{1-2a\cos\left(\frac{2\pi k}{n}\right)+a^{2}}=\frac{n}{1-a^{n}} How do prove this relation?$$\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{ }\...
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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, ...
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### Is it possible to rewrite this sum in terms of some power series?

Is it possible to rewrite this sum in terms of some power series, maybe some cosine power series? $$\sum_{n=0}^{\infty} \dfrac{x^{2n}}{2^{2n}(n!)^2}$$
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### Trying to prove $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$

It has been more than 7 days I have been trying to prove this following result using Harmonic Numbers Let me add this Proving \$\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\...
Hello guys i have a problem evaluating the following sum $$\sum_{n=1}^{+\infty}\frac{n(n+1)}{2}\frac{4x(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ It is obviously of the polygamma ...
I have a problem evaluating the following sum, $$\sum_{n=1}^{+\infty}\frac{4nx(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ The sum obviously is of the form of a polygamma function. ...