Exercise 17. Show that if $\left\{a_n\right\}_{n=0}^{\infty}$ is a sequence of non-zero complex numbers such that \[ \lim _{n \rightarrow \infty} \frac{\left|a_{n+1}\right|}{\left|a_n\right|}=L \] then \[ \lim _{n \rightarrow \infty}\left|a_n\right|^{\frac{1}{n}} . \] In particular, this exercise shows that when applicable, the ratio test can be used to calculate the radius of convergence of a power series.