The infinite series above has also been called the Leibniz series or Gregory–Leibniz series (after the work of James Gregory). It is also called more recently Madhava-Leibniz series, after the discovery that it is a special case of a more general series expansion for the inverse tangent function, first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century. The series for the inverse tangent function, which is also known as Gregory's series, can be given by:
The Leibniz formula for π/4 can be obtained by plugging x = 1 into the above inverse-tangent series.
Comparison of the convergence of the Leibniz formula (cyan squares) and several historical infinite series for π. Sn is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail)
Leibniz's formula converges extremely slowly: it exhibits sublinear convergence. Calculating π to 10 correct decimal places using direct summation of the series requires about five billion terms because for .
If the series is truncated at the right time, the decimal expansion of the approximation will agree with that of π for many more digits, except for isolated digits or digit groups. For example, taking five million terms yields
where the underlined digits are wrong. The errors can in fact be predicted; they are generated by the Euler numbersEn according to the asymptotic formula
where N is an integer divisible by 4. If N is chosen to be a power of ten, each term in the right sum becomes a finite decimal fraction. The formula is a special case of the Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz formula.