Taylor series of function with Lagrange Remainder

Question

Assume function $f : (x_{0} - \varepsilon, x_{0} + \varepsilon) \to \mathbb R , x_0 \in \mathbb R,$ $\varepsilon > 0,$ is n-times differentiable. Define Taylor's series (with Lagrange Remainder) for function $f$ in point $x_0$.

Answer 1

Taylor series with Lagrange Remainder is given by $$f(x)=f(x_0)+f'(x_0)(x-x_0)+f''(x_0)\frac{(x-x_0)^2}{2!}+\cdot\cdot\cdot+f^{(n)}(x_0)\frac{(x-x_0)^n}{n!}+R_n$$

The remainder $R_n$ is given by: $$R_n=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}$$ for some $\xi\in(x_0,x)$.

Refer to Taylor series and Taylor's theorem for further details.