Prove that either $r \geq 1$ or $r = \infty.$

Question

Let $\{a_n \}_{n=0}^{\infty}$ be a sequence of real numbers such that $\sum\limits_{n=0}^{\infty} |a_n|^2 < \infty.$ Consider the power series $\sum\limits_{n=0}^{\infty} a_n x^n$ with radius of convergence $r.$ Show that either $r \geq 1$ or $r = \infty.$

How do I prove that? Any suggestion or hint regarding this will be highly appreciated.

Thank you very much for your valuable time.

Answer 1

You need to prove that $$\sum_{n=0}^\infty a_nz^n$$ converges absolutely when $|z|<1$. I suggest using Cauchy-Schwarz. By hypothesis, the $\ell^2$-norm of the sequence $(a_n)$ is finite.

Answer 2

For any bounded sequence $(a_n)$ the power series $\sum a_n x^{n}$ converges for $|x| < 1$ (by comparison with a geometric series). Hence the radius of convergence is a real number $\geq 1$ or $+\infty$. Note that $\sum |a_n|^{2} <\infty$ implies that $a_n \to 0$ and hence $(a_n)$ is bounded.