Properties of certain sequences related to Stirling's approximation for the gamma function (Q1424729)

The monotonicity properties of certain sequences related to the Stirling formula are studied. Put \(n!= T_n\cdot S_n\), where \(S_n= \sqrt{2\pi n}\cdot({n\over e})^n\). Let \(a_n= 12n- 1/(T_n-1)\) and \(b_n= n({1\over 2}- a_n)\). Then the sequences \((a_n)\) and \((b_n)\) are strictly increasing. The first terms in the asymptotic expansion of \(a_n\) are also given. For example, \(\lim_{n\to\infty} a_n={1\over 2}\), \(\lim_{n\to\infty} b_n= {293\over 720}\).