Ratio Test and Radius of Convergence

The following is an excellent video on how to find radius of convergence using Ratio Test by Khan Academy. Tip: If the speech is too slow, you may want to adjust to 2x speed on YouTube.

The rigorous proof of the Ratio Test requires the formal definition of limit, taught in e.g. MA2108 Mathematical Analysis I.

The rough idea is that $\displaystyle\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|<1$ means that for large $n$ onwards, the series is (less than) a geometric progression with common ratio $|r|<1$ , which is known to converge. Thus the series itself converges.