singularities, poles or essential singularities), which limit the radius of convergence of the series. If the point z = 1 is close to or on the boundary of the disk of convergence, the series for S will converge very slowly. One can then improve the convergence of the series by means of a conformal mapping that moves the singularities such that the point that is mapped to z = 1 ends up deeper in the new disk of convergence. The conformal transform needs to be chosen such that , and one usually chooses a function that has a finite derivative at w = 0. One can assume that without loss of generality, as one can always rescale w to redefine . We then consider the function Since , we have f(1) = g(1). We can obtain the series expansion of g(w) by putting in the series expansion of f(z) because ; the first n terms of the series expansion for f(z) will yield the first n terms of the series expansion for g(w) if . Putting w = 1 in that series expansion will thus yield a series such that if it converges, it will converge to the same value as the original series. Non-linear sequence transformations Examples of such nonlinear sequence transformations are Padé approximants, the Shanks transformation, and Levin-type sequence transformations. Especially nonlinear sequence transformations often provide powerful numerical methods for the summation of divergent series or asymptotic series that arise for instance in perturbation theory, and may be used as highly effective extrapolation methods. Aitken method A simple nonlinear sequence transformation is the Aitken extrapolation or delta-squared method, defined by This transformation is commonly used to improve the rate of convergence of a slowly converging sequence; heuristically, it eliminates the largest part of the absolute error. See also Shanks transformation Minimum polynomial extrapolation Van Wijngaarden transformation References C. Brezinski and M. Redivo Zaglia, Extrapolation Methods. Theory and Practice, North-Holland, 1991. G. A. Baker Jr. and P. Graves-Morris, Padé Approximants, Cambridge U.P., 1996. Herbert H. H. Homeier: Scalar Levin-Type Sequence Transformations, Journal of Computational and Applied Mathematics, vol. 122, no. 1–2, p 81 (2000). , . Brezinski Claude and Redivo-Zaglia Michela : "The genesis and early developments of Aitken's process, Shanks transformation, the -algorithm, and related fixed point methods", Numerical Algorithms, Vol.80, No.1, (2019), pp.11-133. Delahaye J. P. : "Sequence Transformations", Springer-Verlag, Berlin, ISBN 978-3540152835 (1988). Sidi Avram