Some elegant approximations and asymptotic formulas of Ramanujan (Q1184771)

The authors prove three results from the unorganized pages of Ramanujan's second notebook. Two of these results provide the asymptotic expansions of the integral \(\int_ 0^ \infty xe^{-\alpha x^ 2}/(e^{2\pi x}- 1)dx\), as \(\alpha\to 0\), and of the integral \(\int_ a^ \infty (a/x)^ x dx\), as \(a\to\infty\). The last result is concerned with the asymptotic approximation of a certain sum.