Convergence speeding,convergence and summability

Four general principles are proposed for use in constructing convergence speeding methods. Five old methods and about 15 more recent non-linear methods are discussed with reference to their range of applicability and rate of convergence speeding when applied to alternating, monotonic, slightly erratic, Fourier and alternating divergent series. Several of these methods are proved to give analytic continuations, while even asymptotic series with zero radius of convergence can be summed accurately. Divergent series of positive terms may also in some cases be interpreted with moderate accuracy using four different methods. Convergence tests are reviewed in the light of these convergence speeding methods and analytic continuation.