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.
Chen, C-P., & Paris, R. B. (2017). Series representations of the remainders in the expansions for certain functions with applications. Results in Mathematics, 71(3-4), 1443–1457. https://doi.org/10.1007/s00025-016-0612-1