Euler's constant, Stirling's approximation and the Riemann zeta function

Euler's constant 7 is defined to be the limit of the sequence

as Ntends to infinity. This sequence converges rather slowly. We show how to accelerate the convergence by expanding the function f(x) = 1/xin a power series about each integral value x = nthen integrating term by term, and hence expressing the difference between 7 and γ(N) as a power series in 1/N,

Evaluating the terms shown at N= 10 yields γ = 0.577 215 664 9... accurate to ten decimal places. Applying the same technique to f(x)= log_e xwe establish Stirling's series for the factorial function. Approaching the sequence of functions f(x) =1/x ^k =2, 3,... in the same way yields similar series for the Riemann zeta function ζ(k)evaluated at k= 2,3,....