Series Representations of the Remainders in the Expansions for Certain Functions with Applications

Abstract

We present a summary of the series representations of the remainders in the expansions in ascending powers of t of \({2/(e^t+1)}\), sech t and coth t and establish simple bounds for these remainders when \({t > 0}\). Several applications of these expansions are given which enable us to deduce some inequalities and completely monotonic functions associated with the ratio of two gamma functions. In addition, we derive a (presumably new) quadratic recurrence relation for the Bernoulli numbers Bn.