A method of numerically evaluating slowly convergent monotone series is
described. First, we apply a condensation transformation due to Van Wijngaarden
to the original series. This transforms the original monotone series into an
alternating series. In the second step, the convergence of the transformed
series is accelerated with the help of suitable nonlinear sequence
transformations that are known to be particularly powerful for alternating
series. Some theoretical aspects of our approach are discussed. The efficiency,
numerical stability, and wide applicability of the combined
nonlinear-condensation transformation is illustrated by a number of examples.
We discuss the evaluation of special functions close to or on the boundary of
the circle of convergence, even in the vicinity of singularities. We also
consider a series of products of spherical Bessel functions, which serves as a
model for partial wave expansions occurring in quantum electrodynamic bound
state calculations.