We present a summary of the series representations of the remainders in the expansions in ascending powers of t of 2/(et+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.
- Completely monotonic functions
- Bernoulli polynomials and numbers
- Euler polynomials and numbers