Pi, Euler Numbers, and Asymptotic Expansions

Abstract

Gregory’s series for π, truncated at 500,000 terms, gives to forty places $$4\sum\limits_{k = 1}^{500.000} {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{{2k - 1}}} = 3.141590653589793240462643383269502884197$$