Taylor Series

An infinite Series representing a Function, expressed in terms of the function’s derivatives at a single point. It allows complex functions to be approximated by Polynomials.

If a function $f(x)$ is infinitely differentiable at a point $a$, the Taylor series centered at $x=a$ is the following:

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

Expanded, this looks like:

$$f(x)=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\dots$$

Maclaurin Series

A Taylor Series centered at zero ($a=0$), is called a Maclaurin Series

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

Common Series

$$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}x\in \mathbb{R}$$ $$\sin (x)=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}x\in \mathbb{R}$$ $$\cos (x)=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}x\in \mathbb{R}$$ $$\ln (x)=\sum _{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n}(x-1{)}^{n}x\in (0,2]$$