The asymptotic behaviour of algebraic approximants

We study the convergence of algebraic approximants to a function represente d by a power series. We consider, for an arbitrary but fixed degree, approx imant sequences which can be generated recursively by use of Sergeyev's alg orithm. For the exponential function, a logarithmic function and a power of a binomial, we find explicit formulae for the coefficients that appear in a resulting linear recurrence relation. We assume that the error equation m ay be linearized for small errors. Analysis then yields the generic dominan t term in the asymptotic behaviour of the error when a large number of term s of the series are used. Extensive numerical results confirm the behaviour . Finally, we compare this behaviour with that for the closely related meth od of Drazin & Tourigny, in which the degree of the approximants grows with out bound.