What could be a good example for: $\frac{\rm d}{{\rm d}x}\sum_{n\in\mathbb N}f_n(x)\neq \sum_{n\in\mathbb N}\frac{\rm d}{{\rm d}x}f_n(x)$?

Question

I need an power series example for which $$\frac{\rm d}{{\rm d}x}\sum_{n\in\mathbb N}f_n(x)\neq \sum_{n\in\mathbb N}\frac{\rm d}{{\rm d}x}f_n(x)$$

here $f_n(x)$ is differentiable for all $n \in \Bbb N$

Answer 1

Take $$-\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^nx^n}{n}$$ at $x=1$.

$$\left.\frac{d}{dx}\right|_{x=1}\left(\sum_{n=1}^\infty \frac{(-1)^nx^n}{n}\right)=-\frac{1}{2},$$ but $$\left(\left.\sum_{n=1}^\infty \frac{d}{dx}\frac{(-1)^n}{n}x^n\right)\right|_{x=1}$$ doesn't exist.