Flint Hills Series


The Flint Hills series is the series


(Pickover 2002, p. 59). It is not known if this series converges, since csc^2n can have sporadic large values. The plots above show its behavior up to n=10^4. The positive integer values of n giving incrementally largest values of |cscn| are given by 1, 3, 22, 333, 355, 103993, ... (OEIS A046947), which are precisely the numerators of the convergents of pi, 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 pi, and in particular, convergence would imply mu(pi)<=2.5, which is much stronger than the best currently known upper bound.

See also

Cookson Hills Series, Irrationality Measure, Tanc Function

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Alekseyev, M. A. "On Convergence of the Flint Hills Series." 27 Apr 2011.Pickover, C. A. "Flint Hills Series." Ch. 25 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 57-59 and 265-268, 2002.Sloane, N. J. A. Sequence A046947 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Flint Hills Series

Cite this as:

Weisstein, Eric W. "Flint Hills Series." From MathWorld--A Wolfram Web Resource.

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