ABSTRACT
This chapter discusses convergence of formal power series, and considers the implications of writing coefficients as complex contour integrals. It illustrates very classic aspects of series analysis. The chapter describes very precise asymptotic estimates about counting sequences by viewing formal series as analytic functions. The growth of the coefficients of a series is directly related to how the function behaves at its singularities. There are some combinatorial arguments for finding the dominant singularities that might be applicable to combinatorial generating functions. First-order asymptotics are typically given by the dominant singularity, and the others are often safely ignored as their effect on the asymptotics decays exponentially.
