93.51 A 'Sterling' summation method

摘要

A good exercise in [1] asks second semester calculus students toevaluate the series Tk =for k = 1, 2 and 3. Substantial hints aren=in!given for the cases k = 2 and 3. In [2], the value of Tk is listed for thevalues of k from 1 to 8 without a proof (the series converges for all integersk by the ratio test, for instance). In this note, we will provide an approachthat can be used, among other things, to evaluate Tk where k is any non-negative integer. The approach is based on the concept of Stirling numbers.