ABSTRACT
In this chapter, the authors change a dominant theme in our music since it is precisely the question of convergence, including analyticity, and the location and type of singularities, that play a leading role in deducing asymptotic information from generating functions. This musical change will provide some sweet harmonies. It presents some statements without proofs (formulated as theorems) from advanced calculus and the theory of power series. A more precise expression of the radius of convergence is Cauchy's nth-root convergence test. The generating function approach is a powerful technique for solving recurrences. However, extracting coefficients from a corresponding generating function may not be possible in many cases. The generating function approach is a powerful technique for solving recurrences. The chapter presents many interesting examples and problems to solve that show the diversity of asymptotic methods.
