Let $\sum_{n=1}^{\infty} a_{n} x^{n}$ be a real power series that diverges for at least one value of $x$. Show that there exists a non-negative real number $R$ such that $\sum_{n=1}^{\infty} a_{n} x^{n}$ converges absolutely whenever $|x|<R$ and diverges whenever $|x|>R$.

Find, with justification, such a number $R$ for each of the following real power series:

(i) $\sum_{n=1}^{\infty} \frac{x^{n}}{3^{n}}$;

(ii) $\sum_{n=1}^{\infty} x^{n}\left(1+\frac{1}{n}\right)^{n}$.