I am really sorry if this question sounds stupid or too obvious. Even though I know how to apply all of these when calculating limits or sums of series, I still have questions that I need to answer.
1) Taylor's theorem says that a k-times differentiable function can be approximated by a k-th order Taylor's polynomial in the neighborhood of some given point.
Okay, this seems absolutely clear. We have a function and we want to linearize this function near some $a$.
2) Taylor's series is a representation of a function that is infinitely differentiable at a real or complex number a.
Should we say this representation is true in the neighborhood of $a$?
Let's say we want to express $e^x$ as a power series (at $a=0$). Do we need to say that this power series holds in the neighborhood of zero? Or just at $0$?
3) If I understand correctly, a power series is technically the same as a Taylor series, but most of the time we are interested in using Taylor's series. Is that correct?
Can you please clarify the questions above for me?