Radius of convergence proof (sequence of series)

Question

In the proof of the theorem, it says if $|z|>R$ then a similar argument proves that there exists a sequence of terms in the series whose absolute value goes to infinity. What does it mean? How should I do that?

Answer 1

If $\lvert z\rvert>R$, choose $\varepsilon>0$ such that$$(L-\varepsilon)\lvert z\rvert=r>1.$$By the definition of $L$, we have $\lvert a_n\rvert^{1/n}\geqslant L-\varepsilon$ infinitely often. But, when this happens, we have$$\lvert a_n\rvert\lvert z^n\rvert\geqslant\bigl((L-\varepsilon)\lvert z\rvert\bigr)^n=r^n.$$Since $r>1$…