Sequence transformations are methods that manipulate sequences and series to accelerate their convergence.
Consider the convergent series for π/4 and Wynn's celebrated ϵ-algorithm:
using SequenceTransformations a = Sequence(i->(-1)^(i-1)/(2i-1)) s = cumsum(a) ϵ(s)(1:20)-π/4
Certain sequence transformations are powerful enough to sum divergent series. Consider the Euler series:
Gompertz = -.596347362323194074341078499369279376074 a = Sequence(i->(-1)^i*gamma(i)) s = cumsum(a) Levin(s)(1:20)-Gompertz Weniger(s)(1:20)-Gompertz
E. J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Rep., 10:189--371, 1989.
14 days ago