Suppose we have a series of functions $$\sum_{x=0}^\infty h(\theta, x)$$
The question is: is it correct that $$\partial _\theta \sum_{x=0}^\infty h(\theta, x)=\sum_{x=0}^\infty \partial _\theta h(\theta, x)$$
I would assume that the following arguments are correct: Take first the sequence not to $\infty$ but to $n$ $$\partial _\theta \sum_{x=0}^n h(\theta, x)=\sum_{x=0}^n \partial _\theta h(\theta, x)$$
We already know that this holds. Now simply take the limit as $n\to \infty$. We see then that the desired result is correct, so long as for each $x\in \mathbb N$, $\partial _\theta h(\theta, x)$ exists, and if the sequence $\sum_{x=0}^n \partial _\theta h(\theta, x)$ converges.
However, my textbook says that we need additional assumptions. We need $\partial _\theta h(\theta, x)$ to be continuous, and we need the series $\sum_{x=0}^n \partial _\theta h(\theta, x)$ to "converge uniformly".
Why do we need these additional assumptions and what is wrong with my (sketchy) proof?
