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.