Many differential equations cannot be solved analytically using elementary functions, necessitating approximate methods such as a series approach. As a powerful tool for obtaining approximate solutions, this approach forms the core of this book in solving ordinary differential equations. Divided into six chapters, the book defines a specific topic in each, outlines the properties of the method, and works through illustrative examples. Chapter 1 covers Taylor and Maclaurin series, providing the foundation for the subsequent chapters. Chapter 2 details power series solutions to linear ordinary differential equations, while Chapter 3 introduces the Taylor series method for solving them. Chapters 4 and 5 extend both approaches to boundary value problems; Chapter 4 focuses on the Taylor series ansatz, whereas Chapter 5 utilizes the more general power series ansatz. Finally, Chapter 6 presents Picard's method of successive approximations for solving initial value problems. This book is intended for students, lecturers, and all readers in mathematics.