Ratio Test on series with negative powers of $x$

by infinitylord   Last Updated January 12, 2018 22:20 PM

I just found an Asymptotic expansion of the error function as $x \to \infty$ to be:,

$$\text{erf(x)} \approx \frac{2}{\sqrt{\pi}} + \frac{e^{-x^2}}{\sqrt{\pi}x} \sum_{n = 0}^\infty \frac{(-1)^{n+1} (2n-1)!!}{(2x)^n}$$

I wanted to find the radius of convergence, which I believe to be $0$.

I know I can use a variation of the root test on series with reciprocal powers (as I've used before in Laurent series), but this seems much more difficult than using the ratio test here. However, I am unsure if or how the ratio test carries over. I see no reason to believe it wouldn't work, and it likely works much the same way as with ordinary power series.

Alternatively, I thought I may still be able to cite the fact that for $\sum_n a_n$ to converge, it must satisfy $\lim_{n \to \infty} a_n = 0$. In which case I can show that for any fixed $x$, the argument of the sum approaches infinity, and thus it is divergent everywhere.