= Solution
Suppose the <series> converges at $w\ne0$. Then its terms $a_nw^n$ are bounded, say by $M$. Whenever $|z|<|w|$,
$$
|a_nz^n|
\leq M\left|\frac zw\right|^n,
$$
so comparison with a geometric <series> proves absolute convergence at $z$.
Let
$$
R=\sup\{|w|:\sum_{n=0}^{\infty}a_nw^n
\text{ converges}\},
$$
with the natural values $0$ or $\infty$ allowed. If $|z|<R$, choose a convergent point $w$ with $|z|<|w|\leq R$; the preceding argument gives convergence at $z$. If $|z|>R$, convergence at $z$ would contradict the definition of $R$. Hence the <series> converges for $|z|<R$ and diverges for $|z|>R$. No universal assertion is possible on $|z|=R$.
Back to article page
