ca.analysis and odes - What functions are not represented by their power series?

Functions are represented by their power series if and only if they are analytic, i.e. complex differentiable.

Power series are easier to understand as functions of complex variables. For example, there's no apparent reason why the power series of 1/(1 + x^2) centered at 0 should have radius of convergence 1. It's infinitely differentiable everywhere. But as a function of a complex variable, it has a singularity at x = i, and that's why the radius of convergence is 1.