Proving limit of a sequence without Taylor series

Question

I'm trying to prove that pi is irrational with Niven's proof. In the proof I need to show the following: $lim_{n\to\infty}\frac{\pi^{n+1}a^n}{n!}=0$ where $\pi=\frac{a}{b}$ to obtain the contradiction.

I know that this can be done by writing a Taylor series that converges meaning that the terms must go to 0, however, I would like to find a way around that using an $\epsilon - N$ proof. Any help will be appreciated.

Answer 1

So your sequence is $\;a_n:=\cfrac{a^{2n+1}}{b^{n+1}n!}\;$ . Look at the numerical series with this general term, and since it is a positive series apply the $\;n\,-$th root test to it:

$$\sqrt[n]{\frac{a^{2n+1}}{b^{n+1}n!}}=\frac{a^2}{b}\sqrt[n]\frac ab\frac1{\sqrt[n]{n!}}\xrightarrow[n\to\infty]{}\frac{a^2}b\cdot1\cdot0=0$$

and thus the infinite series $\;\sum\limits_{n=1}^\infty a_n\;$ converges $\;\implies \lim\limits_{n\to\infty}a_n=0\;$