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 O((log N)2
M(N)), where M(N) is the time needed to multiply two N-bit numbers. Applications include fast algorithms for elementary functions, π, hypergeometric functions at rational points,
ζ(3), Euler's, Catalan's and Apéry's constant. The algorithms are suitable for parallel computation.