Explain what is meant by the radius of convergence of a power series.

Find the radius of convergence $R$ of each of the following power series: (i) $\sum_{n=1}^{\infty} n^{-2} z^{n}$, (ii) $\sum_{n=1}^{\infty}\left(n+\frac{1}{2^{n}}\right) z^{n}$.

In each case, determine whether the series converges on the circle $|z|=R$.