THEOREM 17.5 .2 Termwise Differentiation of Series
Let $\sum_{n=1}^{\infty} a_{n}(x)$ converge on an $x$ interval $I$. Then
$$
\frac{d}{d x} \sum_{n=1}^{\infty} a_{n}(x)=\sum_{n=1}^{\infty} \frac{d}{d x} a_{n}(x)
$$
if the series on the right converges uniformly on $I$. (The theorem is from the book "Engineering Mathematics, M.D. Greenberg 2nd edition, page 875.)
My question is why uniform convergence allows us to have such equality? Initially I thought distributivity allows such equality needs to hold but I couldn't understand why uniform convergence have such superiority. Also, could you give an example of such series that holds the given equality but not uniformly convergent. Thanks in advance.