Taylor series

A Taylor series is an infinite sum of polynomial terms to approximate a function in the region about a certain point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} . This is only possible if the function is behaving analytically in this neighbourhood. Such series about the pointFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} , as a first approximation we use the value at another point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

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