= Solution
For a nonzero $x$, the absolute successive-term ratio in the exponential series is $|x|/(n+1)\to0$, so the first series converges for every $x$. For the factorial series the ratio is $(n+1)|x|\to\infty$, so its terms fail to tend to zero for every $x\ne0$.
The third series is sparse: its exponent is $n^2$, as in the PDF. If $t_n=(n!)^2|x|^{n^2}$, then
$$
\frac{t_{n+1}}{t_n}=(n+1)^2|x|^{2n+1}\longrightarrow0\qquad(0<|x|<1).
$$
For $|x|>1$, the terms grow without bound; at $|x|=1$ their moduli are $(n!)^2$, so they again fail the term test. Equivalently, the <Cauchy-Hadamard theorem> for the sparse coefficients uses $(n!)^{2/n^2}\to1$, not $(n!)^{2/n}\to\infty$. Thus the three <radii of convergence> are
$$
\boxed{R_1=\infty,\qquad R_2=0,\qquad R_3=1.}
$$
Zeros at non-square coefficient indices do not change the relevant limsup in the last root test. This illustrates the <radius of convergence of a sparse power series>.
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