Taylor and Padé
Function approximation

Let's start with the case where the function f to be approximated is analytic in a disk centered in the middle of the interval on which the approximation is sought, and where the derivatives of f, and therefore its Taylor coefficients, are known at the center (we'll assume without loss of generality that the latter lies at 0):

(we then have a Maclaurin series). We can then evaluate a finite version of this series, a Taylor polynomial...