Limit of Fourier coefficients

Question

Consider the function $f=x^2$ defined over $[-\pi, \pi]$. It is clearly $\mathcal{C}^{\infty}$ so for any $m \geq 0$ I expect $$\lim_{n\rightarrow \infty} n^m a_n = 0$$ where $a_n$ is the $n$th Fourier coefficient of $f$. The $a_n$ are given by something like $$\frac{4}{n^2} (-1)^n $$ Clearly then, choosing $m=2$ I get $$ \lim_{n\rightarrow \infty} n^2 a_n = \lim_{n\rightarrow \infty} 4(-1)^n \neq 0$$ What's going on? If the decay of fourier coefficients does not apply to this function, to what functions does it apply?