= Solution
Write a nonterminating parity series as $y=x^r\sum_{j\geq0}b_jx^{2j}$, where $r=0$ or $1$. Its <recurrence relation> is
$$
\frac{b_{j+1}}{b_j}=\frac{(2j+r)^2-m^2}{(2j+r+2)(2j+r+1)}.
$$
The <ratio test> applied to its nonzero successive terms has limiting absolute ratio $|x|^2$. Hence the series converges absolutely for $|x|<1$ and diverges for $|x|>1$, with
$$
\boxed{R=1\quad\text{for every nonterminating parity series}.}
$$
At $x=\pm1$, the limiting ratio is one, so the basic <ratio test> alone is inconclusive. Retain the next term of the ratio instead:
$$
\left|\frac{b_{j+1}}{b_j}\right|=1-\frac{3}{2j}+O(j^{-2}).
$$
For large $j$ the ratio is positive. Taking logarithms and summing gives $\log|b_j|=-\tfrac32\log j+O(1)$; the $O(j^{-2})$ remainders have a finite sum. Thus $|b_j|\leq Cj^{-3/2}$ eventually. Comparison with the convergent <p-series> proves <absolute convergence> at both endpoints as well:
$$
\boxed{\text{The nonpolynomial solutions' power series converge throughout }[-1,1].}
$$
The <radius of convergence> one is the distance from the expansion point to the nearest <singular points of a differential equation>, $x=\pm1$. Convergence of function values at those points does not make the solutions analytic there; endpoint <derivatives> can diverge. The terminating <Chebyshev polynomial> is an exception with infinite <radius of convergence>. For example, at $m=0$ the other solution is $\arcsin x$, whose endpoint values are finite while its first <derivative> diverges.
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