I know that the taylor approx. of a polynomial centered at 0, if n gets big enough, is just the polynomial itself.
But why do people always say "centered at 0"… wouldn't we also get the polynomial back if we approximated around any other x? Why is 0 so special?
Best Answer
You will still get back the polynomial if you take the Taylor formula of order $n$ centered around any point $a$ , as long as $n \ge \deg$.
Example:
$$t^n = \sum_{k=0}^n \frac{n(n-1) \ldots (n-k+1) a^{n-k}}{k!} (t-a)^k$$