Summation of series of positive terms by condensation transformations.

Abstract:

The condensation transformation, which maps series of positive terms into more conveniently summed alternating series, each term ${v_j}$ of which is itself an infinite series, is discussed with examples. It is shown that for a large class of extremely slowly convergent series (essentially those dominated by the “logarithmic scale") the series defining the terms ${v_j}$ are more easily summed than the original and may in fact be transformed further if desired. Numerical examples reveal the power of the method.