= Terms of a power series are unbounded outside its convergence disc
{title2=$|z|>R\Rightarrow\sup_n|a_nz^n|=\infty$}
Let a <power series> $\sum_{n\geq0}a_nz^n$ have finite <radius of convergence> $R$. If its terms were bounded in <complex modulus> at a point $z$ with $|z|>R$, say $|a_nz^n|\leq M$, choose $w$ with $R<|w|<|z|$. Then
$$
|a_nw^n|\leq M\left(\frac{|w|}{|z|}\right)^n.
$$
The <comparison test for series> with a convergent <geometric series> would give <absolute convergence> at $w$, contradicting its position outside the convergence disc. Hence the terms are unbounded at every such $z$. The argument also works when $R=0$ and gives a stronger obstruction than the usual <term test for divergence>.
Back to article page
