we defined $\exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$
in the proof of the derivative one book starts to take the derivative of each single term i.e. $\dfrac{d(1 + x + x^2/2! +...)}{dx}$ and the result is obvious, however I'm not really sure why we can just take the derivative of an infinite sum? I was told to always be careful when dealing with infinite sums (in my analysis class) - why is this different?
You can swap derivative and sum since the summation terms converge uniformly in all of R. But again you can simply use the definition of limit and solve it that way
The analysis of nonuniform/uniform/point-wise convergence of the derivative/original series is not really needed in this particular case simply because the original is a power series which converges everywhere, and thus the derivative of $e^x$ equals the derivative of the series equals the series of derivatives, which turns out to be the original series and we already know It's value!
In other words, if you assume there is some $x$ which violates the equality of the series of derivatives and $e^x$, then you have a contradiction.
