I'm currently reviewing some elementary analysis (it's been a while), and I recall the tremendous significance of uniform convergence of a sequence of functions.
If memory serves (though I can't find the exact theorem in baby Rudin), the following holds:
Given $A, B \subseteq \mathbb{R}$. Given
\begin{align} (1) \: \:f_k: A \rightarrow B, & \text{ differentiable on }A \\ (2) \: \: F_n : A \rightarrow B & \text{ by } \\ & F_n(x) = \sum_{k=1}^{n}f_k(x)\\ (3)\:\:F: A \rightarrow B & \text{ with } \\ & F(x) = \lim_{n \rightarrow \infty}F_n(x) , \:\forall\: x\in A.\end{align} If $F_n$ converges uniformly to $F$ on A, then $F$ is differentiable on $A$, and
\begin{align} F'(x) = & \sum_{k=1}^{\infty}f_k'(x). \\ \end{align}
I'm looking for an example that illustrates when this doesn't hold; in particular, I'm trying to find $f_k, A,$ and $B$ such that the above $(1), \:(2),$ and $(3)$ hold, however $F_n$ does not converge uniformly to $F$ on $A$, and as a result
\begin{align} F'(x) \neq & \sum_{k=1}^{\infty}f_k'(x). \\ \end{align}
I've tried a few sequences of functions that I know are (and that I can show to be) pointwise convergent and not uniformly so, but I'm struggling to come up with a series of such functions.
Any hints would be greatly appreciated.
Also, this is not homework. It is merely for my own edification.
It is not enough for $F_n(x) = \sum_{k=1}^nf_k(x)$ to converge uniformly to $F(x)$ to ensure that the series is termwise differentiable with
$$F'(x) = \sum_{k=1}^\infty f'_k(x).$$
The basic theorem states that if $F_n(x)$ converges at at least one point in $A$ and $F_n'(x) = \sum_{k=1}^nf'_k(x)$ converges uniformly on $A$ to $G(x)$, then $F_n$ converges uniformly to a differentiable function $F$. In this case $F'(x) = G(x)$ for all $x \in A$.
It is easy to find an example of a series that fails to converge uniformly and is not everywhere differentiable. Simply find a non-uniformly convergent series of differentiable functions that converges to a function with a discontinuity.
Of more interest is a counterexample that presents a uniformly convergent series that fails to be differentiable at some point.
Take $A = [0,\infty)$ and
$$f_n(x) = \begin{cases}x\, ,\,\,\, n= 0 \\ -\frac{x}{(1+nx^2)(1+(n-1)x^2)} \, , \,\,\, n > 0 \end{cases}$$
Using a partial fraction decomposition we obtain a telescoping sum resulting in
$$F_n(x) = x - \sum_{k=1}^n\frac{x}{(1+nx^2)(1+(n-1)x^2)} = \frac{x}{1 + nx^2} $$.
Since $F_n(x)$ attains a maximum value of $1/(2\sqrt{n})$ at $x = 1/\sqrt{n}$, we see that $F_n$ converges uniformly to $F = 0$ and $F'(x) = 0$ on $A$.
However,
$$F'_n(x) = \frac{1 - nx^2}{(1+nx^2)^2} \to \begin{cases}1 \, , \,\,\, x = 0 \\ 0 \, , \,\,\, x >0 \end{cases},$$
and we see $\lim_{n \to \infty} F_n'(0) \neq F'(0)$.