Is there a way to prove $\pi$ is irrational using any of its infinite series?

Solution 1:

In general, you cannot tell if the value of a converging infinite series is rational or irrational without using tools other than basic algebra.

Whilst it is not difficult to prove that, for example, $\ 1+\frac{1}{2}+\frac{1}{4}+\ldots\ $ converges to a rational number using more-or-less algebra only, I believe there are no well-known methods to prove that conditionally convergent series converge to a rational or irrational number, like the ones being talked about with respect to $\ \pi\ $ in the question.

For if there were well-known methods to prove that infinite series like $\ 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots $ is irrational from analysing the series itself/alone, then probably we could use similar methods to show that the series of $\ e+\pi\ $ is rational or irrational. But this result is not known. Therefore, if such a method exists, that is, a method of looking at the series only and determining if it is rational or irrational, then it hasn't been presented/ is not well-known. One possible reason for this difficulty is that rearranging conditionally convergent series can result in a different value of the series, and so using algebraic tricks like for $\ 1+\frac{1}{2}+\frac{1}{4}+\ldots\ $ are not available for use here.

Solution 2:

I am not entirely sure what do you refer to the infinite series expansion of $\frac{\pi}{4}$, but one of the famous and easy proofs using $\frac{\pi}{4}$ could be the following:

Lambert proved that $\pi$ is irrational by first showing that this continued fraction expansion holds: $\tan (x)=\frac{x}{1-\frac{x^{2}}{3-\frac{x^{2}}{5-\frac{x^{2}}{7-\ddots}}}}$

Then Lambert proved that if $x$ is non-zero and rational then this expression must be irrational. Since $\tan (\pi / 4)=1$, it follows that $\pi / 4$ is irrational and thus $\pi$ is also irrational.