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Gregory Series


GregorySeries

The Gregory series is a pi formula found by Gregory and Leibniz and obtained by plugging x=1 into the Leibniz series,

 pi/4=sum_(k=1)^infty((-1)^(k+1))/(2k-1)=1-1/3+1/5-...
(1)

(Wells 1986, p. 50). The formula converges very slowly, but its convergence can be accelerated using certain transformations, in particular

 pi=sum_(k=1)^infty(3^k-1)/(4^k)zeta(k+1),
(2)

where zeta(z) is the Riemann zeta function (Vardi 1991).

Taking the partial series gives the analytic result

 4sum_(k=1)^N((-1)^(k+1))/(2k-1)=pi+(-1)^N[psi_0(1/4+1/2N)-psi_0(3/4+1/2N)].
(3)

Rather amazingly, expanding about infinity gives the series

 4sum_(k=1)^N((-1)^(k+1))/(2k-1)=pi-(-1)^Nsum_(k=0)^infty(E_(2k))/(4^kN^(2k+1))
(4)

(Borwein and Bailey 2003, p. 50), where E_n is an Euler number. This means that truncating the Gregory series at half a large power of 10 can give a decimal expansion for pi whose decimal digits are largely correct, but where wrong digits occur with precise regularity. For example, taking N=5×10^6 gives

GregorySeriesDigits

where the sequence of differences is precisely twice the Euler (secant) numbers. In fact, just this pattern of digits was observed by J. R. North in 1988 before the closed form of the truncated series was known (Borwein and Bailey 2003, p. 49; Borwein et al. 2004, p. 29).


See also

Gregory's Formula, Leibniz Series, Machin's Formula, Machin-Like Formulas, Mercator Series, Pi Formulas

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References

Borwein, J. and Bailey, D. "A Curious Anomaly in the Gregory Series." §2.2 in Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 48-50, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. "Gregory's Series Reexamined." §1.8.1 in Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 28-30, 2004.Borwein, J. M.; Borwein, P. B.; and Dilcher, K. "Pi, Euler Numbers, and Asymptotic Expansions." Amer. Math. Monthly 96, 681-687, 1989.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 157-158, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 50, 1986.

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Gregory Series

Cite this as:

Weisstein, Eric W. "Gregory Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GregorySeries.html

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