Series approximations of the polylogarithm with applications to Euler sums and certain integrals

A series approximation of the polylogarithm based on values of the Riemann zeta function is proposed, and truncation of this series yields finite approximations of the polylogarithm. Both the series representation and its truncated versions of the series are applied to compute alternating Euler sums and certain integrals such as the Bose-Einstein integral. These approximations are rapidly converging as the n th summand of the respective series decreases at exponential rate λ ⁻ⁿ λ - n with λ ≈ 82 λ ≈ 82 as n tends to infinity. This improves some conventional representations where the n th summand would decrease only at rate 2⁻ⁿ 2 - n .