Challenge Level

In general, any function whose derivatives of all orders are finite can be written as a power series

$$f(x) = f(0) + x f'(0) + \frac{x^2}{2!}f''(0) + \dots + \frac{x^n}{n!}f^{(n)}(0)+\dots$$

This expansion is an infinite series (not a polynomial).
Truncating this series at a given point provides us with a
polynomial approximation to f(x).

The question of how big the errors are in this approximation
is a difficult one to answer, and more details will be discovered
at university in Numerical Analysis courses.