Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 79 / 1 / b

The complete integral is $\Gamma (1/2)=\sqrt{\pi }$. Therefore apply asymptotic expansion by repeated integration by parts to the tail $I(x)={\int }_x^{\infty }{t}^{-1/2}{e}^{-t}dt$:The last integral is bounded by $x^{-7/2}{e}^{-x}$. ThusIn particular the first three additive terms are $\sqrt{\pi }$, $-e^{-x}{x}^{-1/2}$ and $\frac{1}{2}{e}^{-x}{x}^{-3/2}$; the formula also supplies three successive terms of the exponentially small tail if the constant is counted separately. A power expansion alone would retain only $\sqrt{\pi }$ and miss this information.