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User:Constan69

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I am me.


In mathematics, Leibniz' formula for π, due to Gottfried Leibniz, states that

Proof

[edit]

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

Q.E.D.

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.