I am me.
In mathematics, Leibniz' formula for π, due to Gottfried Leibniz, states that
Proof
Consider the infinite geometric series
It is the limit of the truncated geometric series
Splitting the integrand as
and integrating both sides from 0 to 1, we have
Integrating the first integral (over the truncated geometric series ) termwise one obtains in the limit the required sum. The contribution from the second integral vanishes in the limit as
The full integral
on the left-hand side evaluates to arctan(1) − arctan(0) = π/4, which then yields
Remark: An alternative proof of the Leibniz formula can be given with the aid of Abel's theorem applied to the power series (convergent for )
which is obtained integrating the geometric series ( absolutely convergent for )
termwise.