 Solution

For $z\ne 0$,If $R|z|<1$, choose $q$ strictly between that limit and $1$. Eventually $|a_n{z}^n|\le q^n$, proving absolute convergence by the geometric series comparison. If $R|z|>1$, choose $q>1$ below the limit; eventually $|a_n{z}^n|\ge q^n$, so the terms do not tend to zero and the series diverges. The point $z=0$ is harmless. This proves the root test in the needed case and gives