Leibniz formula for π

In mathematics, the Leibniz formula for π, named after Gottfried Leibniz , states that

It is also called Madhava–Leibniz series as 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.

It also is the Dirichlet L-series of the non-principal Dirichlet character of modulus 4 evaluated at s = 1, and therefore the value β(1) of the Dirichlet beta function.

Considering only the integral in the last line, we have:

Therefore, by the squeeze theorem, as n → ∞ we are left with the Leibniz series:

For a more detailed proof, together with the original geometric proof by Leibniz himself, see Leibniz's Formula for Pi.

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 1/2k + 1 < 10⁻¹⁰ for k > 5 × 10⁹ − 1/2.

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