Rather then repeating various arguments, I think it's best to just mention this link, which provides explanations and illuminating examples of the phenomenon of smooth functions with diverging Taylor series: https://mathoverflow.net/questions/72/whats-an-example-of-a-...

Quote by Pietro Majer from the link: "it is more useful to communicate the (historically non-trivial) idea that there is a difference between a function and a representation of it by means of a formula"