Taylor series

Taylor series are an infinite sum of polynomial terms to approximate a function in the region about a certain point $a$. This is only possible is the function is behaving analytically in this neighbourhood. Such series about the point$a=0$ are known as Maclaurin series, after Scottish mathematician Colin Maclaurin. They work by ensuring that the approximate series matches the n^th derivative of the function being approximated when it is approximated by a polynomial of degree n.

Proof

See Taylor's theorem

Series

General formula

$f(x)=f(a)+f'(a)(x-a)+{\frac {f''(a)}{2}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}+\cdots +{\frac {f^{(r)}(a)}{r!}}(x-a)^{r}+\cdots$

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

Exponential & Logarithmic functions

$e^{x}=1+x+{\frac {x^{2})}{2!}}+{\frac {x^{3}}{3!}}+\cdots +{\frac {x^{r}}{r!}}+\cdots \forall x$

$ln(1+x)=x-{\frac {x^{2})}{2}}+{\frac {x^{3}}{3}}-\cdots +(-1)^{r+1}{\frac {x^{r}}{r}}+\cdots (-1<x\leq 1)$