Main Result

S = ∑_{n=0}^{∞} \frac{a(n)}{b(n)} \frac{p(0) ⋯ p(n)}{q(0) ⋯ q(n)}

with a(n), b(n), p(n), q(n) small integers = O(n^{c}) can be computed to N digits in time

O (M(N) (log N)^{2}) = O (N (log N)^{3} log log N).