User:Constan69
From Wikipedia, the free encyclopedia
I am me.
In mathematics, Leibniz' formula for π, due to Gottfried Leibniz, states that
[edit] 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 | x | < 1)
which is obtained integrating the geometric series ( absolutely convergent for | x | < 1)
termwise.
