Taylor's Theorem in One Dimension Constant Version

Theorem

$$f(x_0+h)=f(x_0)+f^{\prime }(x_0)h+\frac{f^{\prime \prime }(x_0)}{2!}{h}^2+\cdots +\frac{f^{(k)}(x_0)}{k!}{h}^k+R_k(x_{0}h)$$

Where:

• $R_k(x_0,h)$ is the remainder satisfying ${\lim }_{h\to 0}\frac{R_k(x_0,h)}{h^k}=0$