On convergence of the Flint Hills series

Abstract

It is not known whether the Flint Hills series ${\sum }_{n=1}^{\infty }\frac{1}{n^3\cdot \sin (n{)}^2}$ converges. We show that this question is closely related to the irrationality measure of $\pi$, denoted $\mu (\pi )$. In particular, convergence of the Flint Hills series would imply $\mu (\pi )\le 2.5$ which is much stronger than the best currently known upper bound $\mu (\pi )\le \mathrm{7.6063...}$. This result easily generalizes to series of the form ${\sum }_{n=1}^{\infty }\frac{1}{n^u\cdot |\sin (n){|}^v}$ where $u,v>0$. We use the currently known bound for $\mu (\pi )$ to derive conditions on $u$ and $v$ that guarantee convergence of such series.

Cited by in corpus (1)

• On the flint hills series