= Solution
For $|z|=r<1$, the inequality $n!\geq n$ for $n\geq1$ gives
$$
\sum_{n=0}^\infty|z|^{n!}
\leq r+\sum_{n=1}^\infty r^n<\infty.
$$
The first $r$ comes from $0!=1$; both initial terms have exponent one. Thus the <factorial-gap power series> converges absolutely inside the unit circle. For $|z|>1$, its terms have magnitudes $|z|^{n!}\to\infty$, so the <term test for divergence> proves divergence and $R=1$.
At every point with $|z|=1$, each term has magnitude one, so the terms cannot tend to zero, regardless of their phases. Hence
$$
\boxed{R=1;\quad\text{divergence at every point of }|z|=1.}
$$
The large gaps between exponents make this a <lacunary power series>; cancellation cannot overcome the necessary condition that the terms of a convergent <series> tend to zero.
Back to article page
