Taylor's Theorem

analysis

Definition

Taylor's Theorem

Let $f$ be $n$-times differentiable on the interval $I=[x_0,x$ (or $[x,x_0]$) and $(n+1)$-times differentiable on the interior of $I$. Then there exists a point $\xi \in I$ such that:

$$f(x)=\sum _{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0{)}^k+\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-x_0{)}^{n+1}$$

The term $R_n(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-x_0{)}^{n+1}$

is called the Lagrange remainder. If $f$ is infinitely often continuously differentiable, then $R_n(x)\to 0$ as $n\to \infty$. In this case, the Taylor series agrees with the function $f(x)$ if ${\lim }_{n\to \infty }{R}_n(x)=0$.

Using this theorem, one can approximate functions that are infinitely often continuously differentiable (and whose derivatives do not grow too rapidly) by polynomials.