Questions tagged [convergence-acceleration]

Use this tag for questions about sequence transformations for improving the rate of convergence of a series.

Convergence acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series or sequence.

Techniques for convergence acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Convergence acceleration can also be used to obtain a variety of identities for special functions; for example, the Euler transform applied to the hypergeometric series yields hypergeometric-series identities.

Convergence acceleration can have interesting theoretical properties as well. For instance, in Apery's proof showing the irrationality of $\zeta(3)$, an important step was rewriting $$ \zeta(3) = \frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^3 \binom{2k}{k}} $$The Ratio Test gives $1/4$ when applied to the new series, compared to $1$ using the usual definition $\zeta(s) = \sum_{n=1}^{\infty} n^{-s},$ $\Re(s)>1$.

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Convergence acceleration technique for $\zeta(4)$ (or $\eta(4)$) via creative telescoping?

Question Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar technique to the ones explained by Alf van der Poorten…
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Generalized limits

Cross-posted to Mathoverflow. $\DeclareMathOperator{\Lim}{Lim}$ $\DeclareMathOperator{\dom}{dom}$ $\DeclareMathOperator{\shift}{\sigma}$ $\DeclareMathOperator{\cesaro}{C}$ After reading Terry Tao's post on generalizations of the limit functional,…
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$35.2850899... $ has a closed form ??

Consider the function $t(x)$ defined as : $$ x_1 = x $$ $$x_2 = x $$ $$ x_3 = 2 x^2 $$ $$ x_4 = 4 x^4 + 2 x^2 $$ and for $n > 4 $ $$ x_{n} = \frac { x_{n-1}^2 + x_{n-2}^2 + x_{n-3}^2}{x_{n-2} + x_{n-3} + x_{n-4} } $$ If the sequence converges to a…
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How to compute this constant with high precision $\sum_{n=1}^\infty \left(\frac{1}{a_n}-\frac{1}{(n+1) \ln (n+1)} \right)$

I'm interested in finding the following constant: $$b=\sum_{n=1}^\infty \left(\frac{1}{a_n}-\frac{1}{(n+1) \ln (n+1)} \right)$$ Where: $$a_1=2$$ $$a_{n+1}=a_n+\log a_n$$ This is related to my recent question where the sequence was first introduced…
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What is the fastest way to $\pi$?

There are many known sequences convergent to $\pi$ with different convergence accelerations. For example both of the following sequences are convergent to $\pi$ when $n$ goes to $\infty$: (a)…
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Value of $\sum_{n=0}^{\infty} \frac{(-1)^n}{\ln(n+2)}$

While testing implementations of Wynn's $\epsilon$-algorithm and Levin's u-transformation I need the value of $$\sum_{n=0}^{\infty} \frac{(-1)^n}{\ln(n+2)} \cdot$$ The results of my algorithms are in agreement with the Pari/GP sumalt value of…
gammatester
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How to accelerate the convergence of $1 + \frac{1}{2^2} + \frac{1}{3^2} + \ldots$?

It is well known that $$ \frac{\pi^2}{6} = 1 + \frac{1}{4} +\frac{1}{9} + \frac{1}{16} + \ldots $$ I am trying to use it to calculate $\pi $. The problem is how to accelerate the convergence of the series on the right hand side. The Shanks…
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Speeding up the convergence of $\zeta(2)$

Let us denote by $S$ the sum of the series $\displaystyle\zeta(2)=1+\frac1{2^2}+\frac1{3^2}+\cdots$ Yes, I know (and you know) that $S=\frac{\pi^2}6$, but that is not relevant for the question that I am about to ask. This series converges slowly. In…
José Carlos Santos
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Can Fourier techniques still be useful in situations where we don't have perfect periodicity (such as a missing point in a lattice)?

If we are dealing with a problem with a periodic component, for example a infinite lattice of particles in one or more dimensions and we want to calculate a solution numerically we can use Fourier techniques such as the Poisson summation formula to…
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Does the Riemann rearrangement theorem affect the Euler summation transform?

I was wondering if the Riemann rearrangement theorem has an affect on the Euler summation transform, since, when applying it to a conditionally convergent series, the rearrangement of terms may affect the value to which the series converges to. On…
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Accelerate convergence of sequence

Is it always possible to extract a subsequence from my generic sequence $(q_n)$, such that the convergence of the subsequence to the same limit $r$ is faster then the original?
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Series acceleration with Fourier-Bessel series coefficients

I was investigating methods for series acceleration and I found this identity: $$e=\sum _{n=0}^{\infty } \left(\sqrt{\frac{\pi }{2}} (2 n+1)\right) I_{n+\frac{1}{2}}(1)$$ where $I$ is the modified Bessel function of the first kind. Could you…
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Pade Approximations convergence acceleration

Why Pade Approximatoins accelerate the convergence of series? Generally speaking, what is there an advantage in the sence of convergence acceleration using rational interpolation? Thanks much in advance!!!
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Accelerated fixed-point for $x=\sin(x)$ convergence rate?

I happened to come up with an idea for accelerating the convergence of fixed-point iteration based on Aitken's delta squared acceleration method. What interests me is the case of $x=\sin(x)$, for which fixed-point iteration is known to give roughly…
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Evaluating a particular sum of products of exponential functions

I'm trying to get a closed-form expression for a particular type of sum, or at least a good way to approximate such sums numerically. I've tried using Mathematica, Maxima, etc, to no avail so far. The simplest example of the type of sum I'm looking…
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