We should note that arctan(1) = π/4. It is the main idea of the proof. We will find a Taylor series representation for the inverse tangent and the proof will be complete.
Finding the series representation
Observe these derivatives at of the inverse tangent at x=0:
From simple observation or mathematical induction, we obtain that the nth derivative is zero if n is even, and it is this when n is odd:
So we can strike out the even derivative terms from the Taylor series. Doing that, we obtain the following series for arctan:
Evaluation at x=1 yields
Hence, it immediately follows that
The above series is very slow to converge, taking about five billion terms to get pi to 10 decimal places.
Product representation for pi
From the Basel problem, it follows that the infinite product representation Euler found for sin(x)/x is, in fact, true; despite it relying on the factoring of an infinite polynomial. This formula is
Since sin(π/2) is equal to 1, it immediately follows that
From here, with a bit of rearranging, we obtain
which gives, when expanded,
Pi is also equal to the values of some definite integrals:
Also this holds:
The first two can be verified via integration, the third one follows from the Weierstrass product for the Gamma function and the fourth one is the result of the Basel problem.