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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?

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    I want to approximate the function in order to sum the approximation. This is where I see the problem: if $x=a(1-\frac{k}{b}$ and set e.g. $a=10,\ b=100$ then for k <90 \ x is greater than 1, for $k \geq 90 \ x$ is greater than 1, and then terms in the asymptotic expansion become larger. Since I need only 1 term (the largest, of course), the problem becomes unbounded.2011-08-22

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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 $

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    If $x$ is less than 1, then you shouldn't be using an asymptotic series; asymptotic series are intended to be used for $x$ of large magnitude...2011-08-22