I know that not every function has a power series expansion. Yet what I don't understand is that for every $C^{\infty}$ functions there is a sequence of polynomial $(P_n)$ such that $P_n$ converges uniformly to $f$. That's to say :
$$\forall x \in [a,b], f(x) = \lim_{n \to \infty} \sum_{k = 0}^{\infty} a_{k,n}x^k$$
But then because it converges uniformly why can't I say that :
$$\forall x \in [a,b], f(x) = \sum_{k = 0}^{\infty} \lim_{n \to \infty} a_{k,n}x^k$$
And so $f$ has a power series expansion with coefficients: $\lim_{n \to \infty} a_{k,n}x^k$.