On the asymptotic representation of the Euler gamma function by Ramanujan

摘要

The problem of approximation to the Euler gamma function on the basis of some Ramanujan's formulas is considered. The function h(x)=(g(x))6−(8x3+4x2+x), where g(x)=(e/x)xΓ(1+x)/π, is studied. It is proved that on the interval (1,∞) the function h(x) is increasing monotonically from h(1)=0.0111976… to h(∞)=1/30=0.0333….