I found one definition/condition in wiki: (I am not familiar with this)

http://en.wikipedia.org/wiki/Taylor_series

http://en.wikipedia.org/wiki/Entire_functionIf f(x) is equal to its Taylor series everywhere it is called entire.

And if you want to exchange the order of limits with the expectation,In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane.

you will need Dominated Convergence Theorem, Monotone Convergence

Theorem, or Fatou's Lemma.