Final answer:
To find the maximum radius of convergence R of power series solutions about the ordinary point x = 0, use the ratio test. If the limit of the absolute value of the ratio of consecutive terms in the power series is less than 1, the series converges. The radius of convergence R is equal to the distance to the nearest singularity of the differential equation.
Step-by-step explanation:
The maximum radius of convergence R of power series solutions about the ordinary point x = 0 can be determined using the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms in the power series is less than 1, then the series converges. Therefore, to find R, we need to find the limit of the absolute value of the ratio of consecutive terms as n approaches infinity. If this limit is equal to 1, then R is the distance to the nearest singularity of the differential equation.