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# Leibniz formula for pi

(This page is not about Leibnitz' formula for the derivative of a product, aka product rule.)
In mathematics, Leibniz' formula for π states that

sum_{n=0}^{infty} frac{(-1)^n}{2n+1} = frac{1}{1} - frac{1}{3} + frac{1}{5} - frac{1}{7} + frac{1}{9} - cdots = frac{pi}{4}.

## Proof

Consider the infinite geometric series

1 - x^2 + x^4 - x^6 + x^8 - cdots = frac{1}{1+x^2}, qquad |x| < 1.
Integrating both sides from 0 to 1, we have

int_{0}^{1} ( 1 - x^2 + x^4 - x^6 + x^8 - cdots ), dx = int_{0}^{1} frac{1} {1+x^2}, dx.
The left-hand side then becomes the required sum, while the right-hand side evalulates to π/4:

frac{1}{1} - frac{1}{3} + frac{1}{5} - frac{1}{7} + frac{1}{9} - cdots = tan^{-1} (1) - tan^{-1} (0) = frac{pi}{4}.
Q.E.D.