Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [summation-method]

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

In mathematical analysis, the need arises to generalize the concept of the sum of a series (limit of a sequence, value of an integral) to include the case where the series (sequence, integral) diverges in the ordinary sense. This generalization usually takes the form of a rule or operation, and is called a summation method.

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Towards a new proof of infinitude of primes ( with possible unified application to other primes of special forms whose Infinitude is unknown):

I'm trying to prove the infinitude of primes as follows: Consider the following partial sum : $$S(p)=\sum_{n=2}^p\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$ The summand is zero for non-primes greater than 5 , and finite and non-decreasing for primes…
bambi
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On proving that $\sum\limits_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24}$

Ramanujan found the following formula: $$\large \sum_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24}$$ I let $e^{2\pi n}-1=\left(e^{\pi n}+1\right)\left(e^{\pi n}-1\right)$ to try partial fraction decomposition and turn the sum into…
Mr Pie
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$\sum_{n=1}^\infty\frac{n}{(2n-1)16^n}\binom{2n}{n}^2\left(\sum_{k=n}^\infty\frac{2^k}{k\binom{2k}{k}}\right)=1-\sqrt2+\log(1+\sqrt2).$

Prove:$$\sum_{n=1}^\infty\frac{n}{(2n-1)16^n}\binom{2n}{n}^2\left(\sum_{k=n}^\infty\frac{2^k}{k\binom{2k}{k}}\right)=1-\sqrt2+\log(1+\sqrt2).$$ I'm sorry that I don't even know how to start. I haven't met this kind of series before. I've learnt…
user765381
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Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?

Regarding the proof by Tony Padilla and Ed Copeland that $1+2+3+\dotsb=-\frac{1}{12}$ popularized by a recent Numberphile video, most seem to agree that the proof is incorrect, or at least, is not sufficiently rigorous. Can the proof be repaired to…
ziggurism
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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. $$ \sum\limits_{k=1}^{n-1} \frac{1}{\sin^2(\frac{k\pi}{n})}\\ \sum\limits_{k=0}^{n-1} \tan(\frac{k\pi}{n})\\ $$ Sum of the reciprocal…
Ethanabc
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Approximation of a summation by an integral

I am going to approximate $\sum_{i=0}^{n-1}(\frac{n}{n-i})^{\frac{1}{\beta -1}}$ by $\int_{0}^{n-1}(\frac{n}{n-x})^{\frac{1}{\beta -1}}dx$, such that $n$ is sufficiently large. Is the above approximation true? If the above approximation is true, by…
Hasan Heydari
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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…
onepound
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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} \sum_{n \geq 1} 1 &= -1/2 \tag{1} \\ \sum_{n \geq…
BigThumb
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Integral form(s) of a general tetration/power tower integral solution: $\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b,cn+d)}$

In many tetration/power tower integrals, one sees a general form of the following. Let this new function be notation used to show the connection between the general result and special cases using types of Incomplete Gamma functions. The goal is to…
Tyma Gaidash
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Summation methods ordered by strength

A summation method is a partial function from scalar sequences to scalars, i.e. an element of the set $\mathbb{C}^\mathbb{N} \rightharpoonup \mathbb{C}$. A summation method $\Sigma_1$ is weaker than a summation method $\Sigma_2$ iff $\Sigma_1…
user76284
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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 null method (most trivial of all, sums every…
Renato Faraone
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If $a_n \to z$, then does $\frac{{n \choose 1} a_1 + {n \choose 2} a_2 \dots {n \choose n} a_n}{2^n} \to z$?

It is well known that $\sum_{k=0}^n{n\choose k} =2^n$. My question: If $z$ is the limit point of an infinite sequence of real numbers $\{ a_n \}$, then does $$\frac{{n \choose 1} a_1 + {n \choose 2} a_2+ \cdots+ {n \choose n} a_n}{2^n}$$ converge to…
Descartes Before the Horse
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How does one get that $1^3+2^3+3^3+4^3+\cdots=\frac{1}{120}$?

While watching interesting mathematics videos, I found one of the papers of Srinivasa Ramanujan to G.H.Hardy in which he had written $1^3+2^3+3^3+4^3+\cdots=\frac{1}{120}$. The problem is that every term on the left is more than $\frac{1}{120}$ yet…
Vidyanshu Mishra
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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…
Darshan P.
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Do we have to use Bernoulli polynomials in the Euler-Maclaurin summation formula?

It may be that I have not picked up the proof, but I cannot see where the third condition of Bernoulli polynomials, given below, is used in the derivation of the Euler-Maclaurin summation formula. The Bernoulli polynomials are defined inductively…
Sputnik
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