radius of convergence of a solution
When you solve a differential equation by a power series, the series is not guaranteed to work for every x — it works only inside a disc around the centre x0. The radius of convergence of a solution is the size of that disc: how far from x0 your series is trustworthy. The beautiful fact is that, for a linear ODE, you can predict this radius before computing a single coefficient.
The theorem (going back to Fuchs and Cauchy) is strikingly clean: for y'' + P(x) y' + Q(x) y = 0 expanded about an ordinary point x0, the power-series solution is guaranteed to converge at least out to the nearest singular point of the equation, measured in the complex plane. So the radius is at least the distance from x0 to the closest singularity of P or Q. You do not need to estimate the coefficients a_k and apply a ratio test; you just locate the singular points (often where the original leading coefficient vanishes) and measure. Convergence may be even better than this in special cases, but this bound always holds.
This matters because it tells you, in advance, where your series approximation can be used and where it will fail. For Legendre's equation about x0 = 0, the singular points are at +1 and -1, so the series converges for |x| < 1 — exactly the physical interval of cos(theta) on a sphere. The 'measured in the complex plane' clause is the subtle part: a singularity you cannot see on the real line (like the poles of 1/(1 + x^2) at +i and -i) still limits a real series, which is why a perfectly smooth-looking real function can have a finite radius of convergence.
For (1 + x^2) y'' + y = 0 expanded about x0 = 0, the standard-form coefficients have singularities where 1 + x^2 = 0, i.e. at x = +i and x = -i. The distance from 0 to either is 1, so the series solution converges for |x| < 1 — even though nothing goes wrong anywhere on the real line.
Invisible complex singularities still cap the radius of a real series solution.
The theorem gives a guaranteed minimum, not the exact radius — a particular solution can converge further if its own coefficients happen to behave better (for instance, a polynomial solution like a Legendre polynomial converges everywhere).