07 - Taylor's Theorem for a function of one variable

Taylor's Theorem for a function of one variable

Suppose $f(x)$ has derivatives on $[a,b]$ up to $(n+1)$th order
For any $x\in (a,b]$ there exists $\xi \in (a,x)$ such that

$$f(x)=f(a)+f^{\prime }(a)(x-a)+\cdots +\frac{f^{(n)}(a)}{n!}(x-a{)}^n+\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-a{)}^{n+1}$$