The Flint Hills series is the series
 |
(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.
