the radius of convergence of a series solution

A power-series solution is only as good as the region where the infinite sum actually converges to a finite value. Outside that region the series is meaningless — its terms grow without bound. The radius of convergence is the distance, measured out from your expansion point, within which the series faithfully represents a genuine solution; it is the reach of your answer.

The beautiful part is that you can usually predict this radius before computing a single coefficient. For y'' + P(x) y' + Q(x) y = 0 expanded about an ordinary point x0, a theorem guarantees that the series solution converges at least out to the nearest singular point of the equation — the nearest place where P or Q misbehaves, measured as a distance in the complex plane. So you find the singular points, measure from x0 to the closest one, and that distance is a guaranteed minimum radius. For example, expanding (1 + x^2) y'' + ... about x0 = 0, the coefficients are singular where 1 + x^2 = 0, that is at x = i and x = -i, each a distance 1 from the origin — so the real series converges at least for |x| < 1, even though nothing visible goes wrong on the real axis.

This is why classifying singular points pays off so directly: they fence in your solution. The radius can be larger than the guaranteed minimum, but never smaller. And it explains a phenomenon that puzzles newcomers — a series for a perfectly smooth-looking real function can suddenly stop converging because of a singularity hiding off in the complex plane, invisible on the real line but very much controlling the radius.