I need to approximate this expression in order to sum it. Asymptotically I obtain $\frac1{\sqrt{\pi}x}+\frac1{2\sqrt{\pi} x^2} + O\left(\frac1{x^3}\right)$. Although this looks fine there is the following problem:
If $x=a(1-\frac{k}{b})$ with $ a< and $\frac{b}{2} \leq k \leq b-1, \ x$ turns out to be greater than 1 for $k < k^{\ast}$ for some $k^{\ast}$ and less than 1 for $k \geq k^{\ast}$. This makes the asymptotic expansion somewhat tricky: for fixed $a,b$ the terms are either growing or contracting and then the whole approximation is essentially wrong. Is there any way to make this asymptotic expansion more exact?