On the asymptotic representation of the Euler gamma function by Ramanujan

The problem of approximation to the Euler gamma function on the basis of some Ramanujanʹs 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… .