I want to compute an approximation of the error function in order to compute probabilities for very extreme events (> 20 standard deviations), and I fail to reconcile results from different sources. The standard power series expansion of the error function for small x is given by (wikipedia):
$$ erf(x) = \frac{2}{\sqrt{\pi}} \sum_{n=1}^{\infty} \frac{(-1)^n x^{2n-1}}{n!(2n+1)}$$
Now according to some sources (eq x12) there is an approximation for large x:
$$ erf(x) \approx 1 - \frac{e^{-x^2}}{\sqrt{\pi}x} \left( 1 - \sum_{n=1}^{\infty} \frac{\prod_{k=1}^{n}2k-1}{(2x^2)^n} \right) $$
Whereas according to other sources (eq 1.15) it's:
$$ erf(x) \approx 1 - \frac{e^{-x^2}}{\sqrt{\pi}x} \left( 1 - \sum_{n=1}^{\infty} (-1)^{n+1} \frac{\prod_{k=1}^{n}2k-1}{(2x^2)^n} \right) $$
Basically, the difference (if I haven't made any mistakes this late at night) is just the flipping signs. Can someone explain to me what I'm missing here, or what the right approximation is?