Transformations to speed the convergence of series
J. Barkley Rosser · Journal of research of the National Bureau of Standards · 1951
Numerical instances are given of the speeding of the convergence of series by the Euler transformation.This is even applied advantageously to certain divergent series, and a rigorous justification is given.An example is given of a series for which use of the Euler transformation is not useful.Instances are given of several less widely known methods.Finally, the method of summation by transformation into a continued fraction is illustrated successfully in the case of certain divergent series.The possibility of applying two different methods in succession to a given series is exploited throughout the paper, in spite of the fact that this often requires summing a divergent series.A remarkably useful such transformation is based on a formula due to Euler:(1)We shall refer to this as Euler's transformation.A purely formal derivation of this is as follows.Recalling that we