= Solution
For $z\ne0$,
$$
|a_nz^n|^{1/n}=|a_n|^{1/n}|z|\longrightarrow R|z|.
$$
If $R|z|<1$, choose $q$ strictly between that limit and $1$. Eventually $|a_nz^n|\le q^n$, proving <absolute convergence> by the <geometric series> comparison. If $R|z|>1$, choose $q>1$ below the limit; eventually $|a_nz^n|\ge q^n$, 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
$$
\boxed{\rho=1/R.}
$$
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