= Solution
For nonzero $b$, the coefficients satisfy $a_n\sim-b/n^2$, so the analytic <power series> has <radius of convergence> $1$. It converges absolutely also at $x=\pm1$, by comparison with $\sum n^{-2}$.
The real <logarithm> in the prescribed solution requires $x>0$. Consequently
$$
\boxed{0<x<1\text{ is the interval of the represented classical solution}.}
$$
The <series> and the function itself have finite limits at $x=1$; in fact the logarithmic companion tends to $1$. Its <derivatives> are singular there, so convergence of its values at the endpoint does not make it a classical solution through that <singular point of a differential equation>. At $x=0$ the logarithmic branch diverges. The independent analytic solution $C(1-x)$ is a <polynomial> and has no such convergence restriction.
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