Stirling's Series Made Easy

(A, B,C, D, . . . are positive constants), where the symbol means that rn lies between any two successive partial sums of the series. Usually (as in [1, pp. 529–534]) this formula is deduced from Euler’s summation formula. In this note we show how to compute in succession the constants A, B,C, . . . by a very simple technique. Only derivatives and limits of ln and of rational functions are involved, plus the following elementary property: if f (x) is stricty decreasing (respectively, increasing) for x > 0 and limx→+∞ f (x) = 0, then f (x) > 0 (respectively, f (x) < 0) for x > 0. In the sequel we call this property simply (P). It also holds, and will be used, for sequences. We also show that the constants obtained by this method are the best ones possible, i.e., they cannot be improved by any method whatsoever. In the second section we adapt our method to obtain Stirling’s series for the logarithm of the gamma function.