Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

The prediction properties of Aitken's iterated Delta^2 process, Wynn's epsilon algorithm, and Brezinski's iterated theta algorithm for (formal) power series are analyzed. As a first step, the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of accuracy-through-order relationships. On the basis of these relationships, the rational approximants can be rewritten as a partial sum plus an appropriate transformation term....

The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular...

We give an apparently new proof of Stirling's original asymptotic formula for the behavior of $\ln z!$ for large $z$. Stirling's original formula is not the formula widely known as "Stirling's formula", which was actually due to De Moivre. We also show by experiment that this old formula is quite effective for numerical evaluation of $\ln z!$ over $\mathbb{C}$, when coupled with the sequence acceleration method known as Levin's $u$-transform. As an homage to Stirling, who app...

We present a method to sum Borel- and Gevrey-summable asymptotic series by matching the series to be summed with a linear combination of asymptotic series of known functions that themselves are scaled versions of a single, appropriate, but otherwise unrestricted, function $\Phi$. Both the scaling and linear coefficients are calculated from Pad\'e approximants of a series transformed from the original series by $\Phi$. We discuss in particular the case that $\Phi$ is (essentia...

The theory of the functional sequences and series is presented; uniformly convergent, convergent in the sense of a mean square and weakly convergent sequences and series are considered. Sequential approach to constructing generalized methods of series summarizing and generalized solutions of the problems of mathematical physics is developed

An approach is suggested defining effective sums of divergent series in the form of self-similar exponential approximants. The procedure of constructing these approximants from divergent series with arbitrary noninteger powers is developed. The basis of this construction is the self-similar approximation theory. Control functions governing the convergence of exponentially renormalized series are defined from stability and fixed-point conditions and from additional asymptotic ...

In this work, an efficient approximation scheme has been proposed for getting accurate approximate solution of nonlinear partial differential equations with constant or variable coefficients satisfying initial conditions in a series of exponential instead of an algebraic function of independent variables. As a consequence: i) the convergence of the series found to be faster than the same obtained by few other methods and ii) the exact analytic solution can be obtained from th...

Let $\{a_{1}, a_{2},\ldots, a_{n},\ldots\}$ be a sequence of complex numbers which has at most polynomial growth and satisfies an extra assumption. In this paper, inspired by a recent work of Sasane, we give an explanation of the sum $$a_{1}+2a_{2}+3a_{3}+\cdots+na_{n}+\cdots,$$ and more generally, for any $k\in\mathbb{N},$ the sum $$1^{k}a_{1}+2^{k}a_{2}+3^{k}a_{3}+\cdots+n^{k}a_{n}+\cdots,$$ from the viewpoint of distributions. As applications, we explain the following summ...

The aim of this paper is to construct a new expansion of $(1+1/x)^x$ related to Carleman's inequality. Our results extend some results of Yang [Approximations for constant e and their applications J. Math. Anal. Appl. 262 (2001) 651-659].

The calculation of one loop integrals at finite temperature requires the evaluation of certain series, which converge very slowly or can even be divergent. Here we review a new method, recently devised by the author, for obtaining accelerated analytical expressions for these series. The fundamental properties of the new series are studied and an application to a physical example is considered. The relevance of the method to other physical problems is also discussed.