Some New Approximations and Proofs for Mills’ Ratio

In this paper, we present some new proofs for asymptotic series of Mills’ ratio. Given these new inequalities, the upper bound of the error of Mills’ ratio can be estimated more exactly. Next, a new double inequality related to this asymptotic series is established. For demonstrating the superiority of our new inequality over the original formula and other classical or recently discovered asymptotic series, some numerical computations are provided. Finally, we give a table of Mills’ ratio.