Taking the partial series gives the analytic result
(3)
Rather amazingly, expanding about infinity gives the series
(4)
(Borwein and Bailey 2003, p. 50), where is an Euler number. This
means that truncating the Gregory series at half a large power of 10 can give a decimal
expansion for
whose decimal digits are largely correct, but where wrong digits occur with precise
regularity. For example, taking gives
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).