I'm finding this topic very confusing and would appreciate some clarification (please keep in mind I'm a student, so I may not understand excessively formal explanations/mathematical notation).
My confusion is twofold:
Is it an approximation or exact?
We talk about Taylor series approximations, yet most definitions of a Taylor series I've seen use an equality:
$f(x) = \sum_{n=0}^\infty \frac{f^{n}(a)(x-a)^n}{n!}$
Which one is it? Why do we use the equality sign above if we're approximating the function?
Why do we take the sum to infinity instead of stopping at n?
Let $p_{n, a}(x)$ be the nth-degree polynomial centered at $a$ that approximates $f(x)$. This function has the same value as $f$ at the point $a$: $f(a) = p_{n,a}(a)$. Furthermore, it has the same derivative as the function all the way up to the $n$-th derivative: $f^{n}(a) = p_{n,a}^{n}(a)$.
Why, then, do we take the sum from $n = 0$ to $n = \infty$ if the $n + 1$ and beyond derivatives are no longer equal?