= Solution
For a nonterminating parity series, consecutive nonzero terms satisfy
$$
\left|\frac{a_{j+2}x^{j+2}}{a_jx^j}\right|=\left|\frac{j^2-n^2}{(j+2)(j+1)}\right||x|^2\longrightarrow|x|^2.
$$
The <ratio test> therefore gives $\boxed{R=1}$ for each nonterminating series. The nearest singular coefficient points are $x=\pm1$, both distance one from the expansion centre.
The polynomial exception is essential here. If $n$ is an integer, the series of the same parity as $|n|$ terminates and has $\boxed{R=\infty}$; the opposite-parity independent series remains nonterminating with radius one. For noninteger $n$, both have radius one. A <regular singular point> can be removable for one particular solution, so its presence does not force every individual solution to have finite convergence radius.
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