State and prove Gregory's series
Answers
Answer:
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. 2
Answer:
Gregory's series is a Taylor series extension of the inverse tangent function with an unlimited number of steps. James Gregory first found it in 1668. Gottfried Leibniz rediscovered it a few years later, obtaining the Leibniz formula for as the particular case x = 1 of the Gregory series.
Step-by-step explanation:
Gregory and Leibniz discovered the Gregory series, which is produced by inserting 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 difference sequence is twice as long as the Euler (secant) numbers. J. R. North noticed this pattern of digits in 1988, long before the closed version of the truncated series was discovered (Borwein and Bailey 2003, p. 49; Borwein et al. 2004, p. 2