Some new results related to the constant e and function (1+1/x)x

Abstract

It is known that the constant e has the following series representations: e = ?? k=0 1/k! and e = ?? k=0 9k2+1/(3k)!. The second series is extremely rapidly convergent. In this paper, we present asymptotic expansions and two-sided inequalities for the remainders Rn and Rn, where Rn = e ? ?n k=0 1/k! and Rn=e??n k=0 9k2+1/(3k)!. Also, we present some inequalities and completely monotonic functions involving (1 + 1/x)x. We also consider a number of related developments on the subject of this paper.