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?
Best Answer
One property of any power series is its "radius of convergence." The Taylor series will converge to the value of the function within the radius of convergence (at least, this will be true for most "normal" functions you'd encounter in an elementary calculus class). Different series will have different radii of convergence.
In the case of $e^x$, you can show that the associated Taylor series converges on the entire real line (i.e., the radius of convergence is infinite), so the Taylor series "holds" (i.e., converges to the exact value $e^x$) everywhere. (Although convergence will be faster near $0$, which is why you'll want to use the Taylor polynomial approximations only near $0$.)
A Taylor series is a type of power series. A power series is just some series where the $n^{th}$ term has an $x^n$ involved. A Taylor series is a series of that type, with the coefficients set by the derivatives of some function.