Taylor series

A Taylor series is an infinite sum of polynomial terms to approximate a function in the region about a certain point $a$. This is only possible if 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 up to 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

An intuitive explanation of the Taylor series is that, in order to approximate the value of $f(x)$, as a first approximation we use the value at another point $a$, i.e. $f(a)$. If $x$ and $a$ are close together and $f$ varies only slowly, this can be a good approximation. Then we refine the approximation step by step. The derivative of $f$ is used to calculate approximately how much $f$ would be expected to change between $a$ and $x$, and this amount is added as a correction. But we assume we only know the derivative of $f$ at $a$, and the derivative may change between the two numbers, so another correction is needed, involving the second derivative which is a measure of how much the first derivative changes. So it continues, adding corrections to corrections, and in the limit, if it converges then it converges to the actual value of $f(x)$ even if $x$ and $a$ are far apart.

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

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

where $\scriptstyle f'$ is the first derivative of the function $\scriptstyle f$, and $\scriptstyle f''$ is the second derivative, and so on.

Exponential & Logarithmic functions

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

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

Trigonometric functions

$\sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots +(-1)^{r}{\frac {x^{2r+1}}{(2r+1)!}}+\cdots \qquad \forall x$

$\cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+\cdots +(-1)^{r}{\frac {x^{2r}}{(2r)!}}+\cdots \qquad \forall x$

$\tan x=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+\cdots +{\frac {B_{2r}(-4)^{r}(1-4^{r})}{(2r)!}}x^{2r-1}+\cdots \qquad |x|<{\frac {\pi }{2}}$
where B_k=k^th Bernoulli number.

Inverse trigonometric functions

$\operatorname {tan^{-1}} x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots +(-1)^{r}{\frac {x^{2r+1}}{(2r+1)}}+\cdots \qquad (-1<x\leq 1)$

Hyperbolic functions

$\sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+\cdots +{\frac {x^{2r+1}}{(2r+1)!}}+\cdots \qquad \forall x$

$\cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+\cdots +{\frac {x^{2r}}{(2r)!}}+\cdots \qquad \forall x$

Inverse hyperbolic functions

$\operatorname {tanh^{-1}} x=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+\cdots +{\frac {x^{2r+1}}{(2r+1)}}+\cdots \qquad (-1<x\leq 1)$