Pi, Euler numbers and asymptotic expansions

Gregory's series for π, truncated at 500,000 terms, gives to forty places 3.141590653589793240462643383269502884197. This is not π to forty places. As one would expect, the 6th digit after the decimal point is wrong. The suprise is that the next 10 digits are correct. In fact, only 4 digits aren't correct. The point of this article is to provide an explanation for this.