Flint Hills Series
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.
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harmonic series