The Taylor Expansion

Taylor Series

$$f(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\dots$$ $$\boxed{f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n}$$

where,

• $x\to$ domain
• $a\in x\to$ center of the series i.e. the point around which the Taylor approximation holds.

Maclaurin Series $a=0$

Point of approximation is the origin

$$\boxed{f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(0)}{n!}{x}^n}$$

Taylor Expansion of $f(x+h)$

Function to be evaluated at $(x+h)$. Center of the series $a=x$

$$\begin{array}{rl}f(x+h)& =f(x)+f^{\prime }(x)((x+h)-x)+\frac{f^{\prime \prime }(x)}{2!}((x+h)-x{)}^2+\dots \\ f(x+h)& =f(x)+f^{\prime }(x)\cdot h+\frac{f^{\prime \prime }(x)}{2!}\cdot h^2+\dots \end{array}$$ $$\boxed{f(x+h)=\sum _{n=0}^{\infty }\frac{f^{(n)}(x)}{n!}\cdot h^n}$$