We describe two techniques for fast multiple-precision evaluation of linearly convergent series, including power series and Ramanujan series. The computation time for N bits is Of(logN)^2M(N)), where M(N) is the time needed to multiply two iV-bit numbers. Applications include fast algorithms for elementary functions, t, hypergeometric functions at rational points, C(3), Euler's, Catalan's and Apery's constant. The algorithms are suitable for parallel computation.