Taylor Series

analysis

Definition

Taylor Series

Let $f$ be an infinitely times differentiable function. The series

$$\sum _{n\ge 0}\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n=f(x_0)+f^{\prime }(x_0)(x-x_0)+\frac{f^{\prime \prime }(x_0)}{2}(x-x_0{)}^2+\dots$$

is called Taylor series of $f(x)$ at expansion point $x_0$. If $x_0=0$, the Taylor series is called MacLaurin series.