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An application: a series for PI

The integrand in the integral formula for arctan, , can be expanded in a geometric series, which can be integrated term by term to get a series formula for arctan y:

Integrating term by term we get

(Integrating term by term is ok if the resulting series is absolutely convergent, which this series is when |y| < 1.)

We can use this series to obtain a formula for .

The tangent of  which means that :

This gives us