We have a power series $\sum a_n x^n$ with radius of convergence $R$. All of the coefficients are integers and an infinite number are not zero. We need to prove that $\lim \sup | a_n | > 0 \implies R \leq 1$.
I am having difficulty proving this fact. Since all of the $a_n$ are integer, this suggests that the $\lim \sup$ is also an integer, and so $\lim \sup | a_n | \geq 1$. It seems, however, that this could grow without bound as the series isn't bounded above, so I do not understand how we would go about finding such a narrow radius of convergence. The easiest way to find $R$ seems to be the root test, and so we would want to consider the amended sequence of coefficients, $|a_n|^{1/n}$. But, I am confused as to how to get to this point.
Any helpful insights would be greatly appreciated.
Since $\sqrt[n]{\lvert a_n\rvert}\geqslant1$ for infinitely many $n$'s, $\limsup_n\sqrt[n]{\lvert a_n\rvert}\geqslant1$. So$$R=\frac1{\limsup_n\sqrt[n]{\lvert a_n\rvert}}\leqslant1.$$