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.
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