I read the following paragraph which claims to be the Taylor expansion of standard normal CDF for positive $x$.
$1 - \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} (a_1k + a_2k^2 + a_3k^3 + a_4k^4 + a_5k^5)$,
where $k:=\frac{1}{1+0.2316419x}, a_1=0.319381530, a_2=-0.356563782, a_3=1.781477937, a_4=-1.821255978, a_5=1.330274429.$
I do not get how this is derived even though I am numerically convinced of its truth. Could anyone explain this to me, please?
This is no Taylor expansion. It is a rational approximation listed as formula 26.2.17 in Abramowitz / Stegun (see http://people.math.sfu.ca/~cbm/aands/page_932.htm) and attributed to Hastings (1955), Approximations for Digital Computers.