Leibniz formula for π

The Leibniz series is a formula to approximate the circle number that developed Gottfried Wilhelm Leibniz in 1682. It reads:

This formula was the Indian mathematician Madhava already known in the 14th century and the Scottish mathematician Gregory before 1671 Leibniz discovered new for the continental European mathematics.

The convergence of these infinite series follows immediately from the Leibniz criterion.

Convergence speed

Is the residual term of the sum by summands

Applies with the error estimate of the Leibniz criterion

More detailed observations even show that

With summands so you can get decimal places with an error < 0.5 in the -th decimal place:

The required number of summands for meaningful decimal places in the result is in accordance with

A list of partial sums, the ' arising from Leibniz formula

Using the Leibniz series it is possible to calculate an approximation of the mathematical constant, because it is

The following list shows the sequence of partial sums of the terms of the sequence multiplied by 4 Leibniz series.

Since the sequence converges very slowly, it is not suitable for efficient calculation of Pi.