I'm trying to refresh my memory on theorems regarding uniform convergence of function series. Specifically, I'm interested in the following Theorem:
Let $f_n : I\longrightarrow\mathbb{R}$ be a sequence of functions defined on an interval $I$. if:
1) $\displaystyle{\sum_{n=1} ^\infty} f_n(x_0)$ converges for some $x_0 \in I$,
2) $f_n(x)$ is differentiable on $I$ for each $n \in \mathbb{N}$
3) $\displaystyle{\sum_{n=1} ^\infty} f_n'(x)$ converges uniformly on $I$
then $\displaystyle{\sum_{n=1} ^\infty} f_n(x)$ converges uniformly on $I$, $\displaystyle{\sum_{n=1} ^\infty} f_n(x)$ is differentiable on $I$ and $\frac{d}{dx} \displaystyle{\sum_{n=1} ^\infty} f_n(x)=\displaystyle{\sum_{n=1} ^\infty} f_n'(x)$ for each $x\in I$
I tried to find what are the restrictions on the interval $I$ (if any). most of the texts I found just write $I=[a,b]$. My questions are:
1) does $I$ have to be closed interval?
2) does $I$ have to be a bounded interval?
3) are there any other generalizations of the above theorem?
Let $I$ be a bounded interval, not necessarily closed. Suppose $\{g_n(x_0)\}$ converges for some $x_0$, $g_n$ is continuously differentiable and $g_n'(x)$ converges uniformly to some function $\phi$. Then $g_n(x)=g_n(x_0)+\int_{x_0}^{x} g_n'(t)\, dt \to \int_{x_0}^{x} \phi (t) \, dt$ uniformly. If $g=\lim g_n$ we get $g(x)=\lim g_n(x_0)+\int_{x_0}^{x} \phi (t)\, dt$. This implies that $g$ is differentiable and its derivative equals $\phi $ (becasue $\phi $ is necessarily continuous). Taking $g_n$ to be the n-th partial sum of the given series we see that the result holds for any bounded interval. For an unbounded interval the only difference is unform convergence of the series must be replaced by uniform convergence on bounded subsets.