Taylor's Formula

analysis

Definition

Taylor's Formula

If a Taylor series is interrupted after $n$ terms, then

$$f(x)=\sum _{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0{)}^k+R_n$$

is given, which is called Taylor’s formula with remainder $R_n$. The sum before the remainder is called the Taylor polynomial of $n$-th order. $R_n$ is the error term and naturally depends on $n$, $x$, and $x_0$.