The Cauchy Ratio test says: If $ \lim_{{n}\to{\infty}}\frac{a_{n+1}}{a_n} < 1 $ then the series converges. OK.

Now I read that for a power series (of functions of x), the same test also provides the interval of convergence, i.e. If the series converges, then $ \lim_{{n}\to{\infty}}\frac{a_{n+1}}{a_n} = {R}^{-1} $ and the interval is -R < x < R

Could someone please explain why this works? Thanks.