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+----- Thread: Flint Hills Series (/showthread.php?tid=2322)
Flint Hills Series - mklabgr - 10-01-2026
The Flint Hills series is the infinite series
$S_1=\displaystyle\sum_{n=1}^{\infty}\frac{\csc^2 n}{n^3}$, introduced by Clifford Pickover. Its unusual feature is the factor $\csc^2 n=1/\sin^2 n$, which can occasionally become extremely large when $n$ is close to a multiple of $\pi$. Although the denominator $n^3$ normally suggests rapid convergence, these sporadic large terms make the convergence of the entire series a difficult and still unresolved question. Numerical plots up to $n=10^4$ illustrate this irregular behavior.
The exceptional values of $n$ are closely connected with the rational approximations to $\pi$. In particular, the integers producing successively larger values of $|\csc n|$ are $1,3,22,333,355,103993,\ldots$, which are the numerators of the convergents in the continued-fraction approximations of $\pi$. At these values, $|\csc n|$ can become enormous—for example, reaching approximately $33173.7$ at $n=103993$. This connection shows that understanding the series requires studying how closely integer multiples can approximate multiples of $\pi$.
A deeper result due to M. Alekseyev (2011) connects the Flint Hills series with the irrationality measure of $\pi$, a quantity describing how well $\pi$ can be approximated by rational numbers. In particular, convergence of the Flint Hills series would imply the strong bound $\mu(\pi)\leq 2.5$, which is substantially stronger than the best currently known upper bound. Thus, the seemingly simple series provides a surprising connection between infinite series, Diophantine approximation, continued fractions, and the arithmetic properties of $\pi$. MathWorld also cites more recent work from 2025 investigating the convergence problem and related bounds.
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