Leibniz formula for pi

Statement

The series $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n + 1}$ converges, and $$1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}.$$

Remarks

The series converges by the alternating series test. Its value comes from the power series of arctangent at $x = 1$, since $\arctan 1 = \pi/4$. The convergence is very slow. The error after the terms up to $n = N$ is at most $1/(2N + 3)$. Mathlib states it as a limit of partial sums, in Real.tendsto_sum_pi_div_four.

History

Madhava of Sangamagrama or his school found the series in the 14th or 15th century. James Gregory found it again in 1671, and Leibniz in 1673.

Explanations

L1 human

Add and subtract the reciprocals of the odd numbers: $1 - 1/3 + 1/5 - \cdots$. The result is a quarter of $\pi$, which links the odd numbers with circles.