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<<b $ 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?
Maple (with MultiSeries) reports this asymptotic for $\mathrm{erfi}(x)$ ... $$ \Biggl(\frac{1}{\sqrt{\pi} x} + \frac{1}{2 \sqrt{\pi} x^{3}} + \frac{3}{4 \sqrt{\pi} x^{5}} + \frac{15}{8 \sqrt{\pi} x^{7}} + O \Bigl(x^{(-8)}\Bigr)\Biggr) \operatorname{e} ^{x^{2}} $$ Maple's definition of erfi is equivalent to: $$ \mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} \operatorname{e} ^{s^{2}} d s $$