In this article we derive all salient properties of analytic functions, including the analytic version of the inverse function theorem, using only the most elementary convergence properties of series. Not even the notion of differentiability is required to do so. Instead, analytical arguments are replaced by combinatorial arguments exhibiting properties of formal power series. Along the way, we show how formal power series can be used to solve combinatorial problems and also derive some results in calculus with a minimum of analytical machinery. (Contains 2 figures.)