(Pickover 2002, p. 59). It is not known if this series converges, since can have sporadic large values.
The plots above show its behavior up to . The positive integer values of giving incrementally largest values of are given by 1, 3, 22, 333, 355, 103993, ... (OEIS A046947), which are precisely the numerators of
the convergents of ,
corresponding to the values 1.1884, 7.08617, 112.978, 113.364, 33173.7, ....

Alekseyev (2011) has shown that the question of the convergence of the Flint Hill series is related to the irrationality measure
of ,
and in particular, convergence would imply , which is much stronger than the best currently
known upper bound.