Let $f_n(x)$ be real functions. Given that the derivatives exist, and that the sums converge, is the following true? If not, what is a counterexample, and when is it true?
$$\frac{d}{dx}\sum_{n=1}^{\infty} f_n(x)=\sum_{n=1}^{\infty} \frac{d}{dx}f_n(x)$$