I've recently started learning about Taylor/Maclauren series and I'm finding it a bit hard to wrap my head around a few things.
So, if $f(x)$ is not infinitely differentiable and we construct a polynomial $p(x)$ such that $p(a) = f(a)$, f'(a) = p'(a), f''(a) = p''(a) etc, for all possible derivatives of $f(x)$, can we say that the functions $f(x)$ and $p(x)$ are equivalent?
Likewise, if $f(x)$ is infinitely differentiable, does the taylor series $p(x)$ become equivalent to $f(x)$ when given an infinite number of terms?