We present in this chapter some concepts and results on power series, which are used in the study of the Frobenius method for solving homogeneous linear second-order ordinary differential equations with nonconstant coefficients. The Frobenius method plays an important role in the solution of this kind of equation, as it will always provide at least one solution, and as we know, another linearly independent solution can in principle be obtained from the first one by reduction of order. At the end, several solved exercises are discussed step by step, specifically the Bessel equation and the Whittaker equation. As a simple application, we discuss a series expansion associated with the displacement of a rest mass particle, resulting from a constant force, including relativistic effects. We conclude with a list of proposed exercises, among which some interesting applications are left to the reader, such as the confluent hypergeometric equation, recurrence relations with three terms, and development around the point at infinity, all of them with the anwser and/or suggestion.