Some functions are not represented by their power series even when they are continuous and have all the necessary derivatives. What's the best characterization of these functions? Explanations at any level are welcome.



What functions are not represented by their power series?
A smooth function is characterized as being analytic if its derivatives
on any closed interval have a certain growth rate. See "Alternate Characterizations"
in the 
[Wikipedia article on analytic functions.](http://en.wikipedia.org/wiki/Analytic_function) 


