Series representations of the remainders in the expansions for certain functions with applications

We present a summary of the series representations of the remainders in the expansions in ascending powers of t of 2/(et+1)2/(et+1) , sech t and coth t and establish simple bounds for these remainders when t>0t>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.

Effects of age and body mass on development of diving capabilities of gray seal pups: costs and benefits of the postweaning fast

Development of adequate diving capabilities is crucial for survival of seal pups and may depend on age and body size. We tracked the diving behavior of 20 gray seal pups during their first 3 mo at sea using satellite relay data loggers. We employed quantile analysis to track upper limits of dive duration and percentage time spent diving, and lower limits of surface intervals. When pups first left the breeding colony, extreme (ninety-fifth percentile) dive duration and percentage time spent diving were positively correlated with age, but not mass, at departure. Extreme dive durations and percentage time spent diving peaked at [Formula: see text] d of age at values comparable with those of adults, but were not sustained. Greater peaks in extreme percentage time spent diving occurred in pups that had higher initial values, were older at their peak, and were heavier at departure. Pups that were smaller and less capable divers when they left the colony improved extreme dive durations and percentage time spent diving more rapidly, once they were at sea. Minimum survival time correlated positively with departure mass. Pups that were heavier at weaning thus benefitted from being both larger and older at departure, but smaller pups faced a trade-off. While age at departure had a positive effect on early dive performance, departure mass impacted on peak percentage time spent diving and longer-term survival. We speculate that once small pups have attained a minimum degree of physiological development to support diving, they would benefit by leaving the colony when younger but larger to maximize limited fuel reserves, rather than undergoing further maturation on land away from potential food resources, because poor divers may be able to "catch up" once at sea.