The first rigorous proof that π is irrational is from Johann Heinri...

That is, assume $\pi$ is rational.

To prove that $f(x)$ and its derivatives $f^{(j)}(x)$ have integral...

Shouldn't that be $F_n(x)$ ?

\begin{eqnarray*} & F''(x) &=f^{2}(x)-f^{4}(x)+f^{6}(x) - ... \\ &...

The idea now is to show that the integral of $f(x) sin(x)$ cannot a...

We can see that $f(x)$ is positive over $(0, \pi)$, and since $\pi$...