## ABSTRACT

This chapter continues the discussion begun in the previous chapter on the algebraic method for finding power series solutions to first- and second-order homogeneous linear differential equations. The main goal here is the verification of the theorems given previously on the existence and radius of convergence of the power series solutions.

The chapter begins by expanding the class of homogeneous linear differential equations under consideration from those with polynomial coefficients to those with coefficients analytic at a given point. This leads to the development of the “reduced forms” for the equations which are then used in proving the desired theorems on the validity of the power series solutions. Along the way, general formulas for the recursion formulas used in the power series solutions are derived. At points, the discussion becomes more technical than in most of the rest of the text, especially in the last half of the chapter covering the details of verifying convergence and in verifying the existence of “the closest singular point” to a given point on the complex plane.