= Solution
For any fixed real $x\ne0$, the ratios between consecutive nonzero terms in the absolute-value series are
$$
\frac{|x|^{2n+2}/(2n+2)!}{|x|^{2n}/(2n)!}=\frac{|x|^2}{(2n+2)(2n+1)}\longrightarrow0,
$$
and
$$
\frac{|x|^{2n+3}/(2n+3)!}{|x|^{2n+1}/(2n+1)!}=\frac{|x|^2}{(2n+3)(2n+2)}\longrightarrow0.
$$
The <ratio test> proves absolute convergence of both <power series> at every such $x$; convergence at zero is immediate. \b[Both <radii of convergence> are therefore $\boxed{\infty}$]. Applying the test to the nonzero terms avoids the undefined coefficient ratios caused by alternate zero coefficients when these are viewed as full <power series> in $x$.
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