Taylor expansion

Taylor series

Definition

Let $I$ be an interval $f\colon I\rightarrow \mathbb{R}$ an arbitrary differentiable function, then the infinite series:

$$ T_f(x) = f(a) + f^{\prime}(a)(x-a) + \frac{f^{\prime \prime}(a)}{2!} (x-a)^2 + ... $$ $$ T_f(x) = \sum_{k=0}^{\infty} \frac{f_{a}^{(k)}}{k!} \left( x-a \right)^{k}$$

is the Taylor expansion of $f$ around the point $a$. For $(x-a)$ being small, one can stop the Taylor expansion after a certain n^th element and obtain thus the approximation of the function $f$ around the point $a$.