# Sufficent condition for existence of power series of Real valued function

I know that for complex valued function if function is holomorphic on some open connected set then there exist convergent power series converging to that function.
But this is not true in case of infinite time differentible Real valued function .Example $e^{-1/x^2}$
I have not encountered yet Sufficent condition for existence of Power series for real valued function.?
I had encountered taylor series but There is residue term is present.So how to say Taylor series as power series of function? Any Help will be appreciated .

There is a simple criterion to conclude the convergence of the power series in some point $x \in I$ towards the function, that is to show that the remainder term in the Taylor approximation vanishes on a neighborhood of $x$. For example, we could use special remainder estimates as the Peano form.
• Here the link to http://quasi-analytic%20function/ is broken. – gammatester Sep 4 '18 at 16:19