Question about Taylor series vs Fourier series I'm starting to study Taylor series for the first time seriously, I kind of ignored them when I first took calculus but now that I'm doing Real Analysis I've been trying to really understand them.
One thing I was curious about is the difference between Fourier and Taylor series (other than that the basis functions are different). What I mean is, if a function meets the condition for a fourier series, then theoretically by taking enough terms, you could re-create the function over the whole real number line just using the series right?
It seems like with taylor polynomials, the idea is that because they are defined in neighborhoods about some point, say $c$, then taking more terms of the series only means that you are re-creating the function in some neighborhood about $c$. But if I took enough terms could I re-create the original function over the whole number line?
Thanks
 A: Fourier series are designed for periodic functions. When you calculate the Fourier series of a function, you assume that this function is periodic. So, when you have a Fouries series that represents a function on the period interval (oftentimes that's $[0,2\pi]$), the Fourier series represents that function on the whole real line.
With Taylor series, it's different. You don't assume that the given function is periodic. You "hope" to express the function in the neighborhood of a given point by its Taylor series centered at that point. An important value for a Taylor series is its "radius of convergence" which says on what part of the real line the Taylor series makes sense. It's not necessarily true that a function defined on all - or most - of $\mathbb R$ can be expressed by its Taylor series centered at a specific point. Consider for example
$$
f(x) = \frac{1}{1-x} = \sum_{j=0}^\infty x^j.
$$
$f$ is defined on $\mathbb R\setminus\{1\},$ and the r.h.s is its Taylor series centered at $0.$ The Taylor series converges for $|x| < 1,$ which doesn't coincide with the domain of definition of $f.$
The behavior of Taylor series becomes much more understandable if you move from $\mathbb R$ to $\mathbb C.$
