= Solution
This is a <lacunary power series>: the coefficient of $z^{n!}$ is $n^n$, while the other coefficients are zero. Along its nonzero coefficients,
$$
(n^n)^{1/n!}=\exp\!\left(\frac{n\log n}{n!}\right)\longrightarrow1.
$$
The factorial dominates $n\log n$; for example $n!\geq n(n-1)(n-2)(n-3)$ for large $n$ already suffices for this limit. Hence the coefficient limsup in the <Cauchy-Hadamard theorem> is one. \b[The <radius of convergence> is] $\boxed{R=1}$.
Directly, for $|z|<1$ the consecutive term ratio is $(n+1)^{n+1}n^{-n}|z|^{n n!}$, which tends to zero; for $|z|\geq1$ the term magnitudes are at least $n^n$ and fail to tend to zero. The factorial is the exponent of $z$, not part of its coefficient.
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