Taylor Series

Ok. There was a physics problem where the movement of a pendulum around a small angle was $R(1-Cos(\theta ))$. And then… Just like magic, the student represented that function as $R(\frac{{\theta }^2}{2})$… WHATTTT???

What are Taylor Series?

They’re a way of representing non-polynomials as polynomials. A key point to note (that’s also pretty cool about them) is that they translate derivative information at a single point to a output around that point.

Taylor Series Formula:

If $f$ has derivatives, of all orders, at $f(a)$, then the Taylor series for the function $f$ at $a$ is

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

Maclaurin Series:

The Taylor series for $f$ at 0 is known as the Maclaurin series for $f$. Pretty scummy to legit get a whole new name for it when Taylor did all the work but ok…

What happens if you extend the Series out into infinity?

Well, if it only ever gets better and better at representing it, then it’s known to converge. Otherwise, if it hits a point on either side of the original point that it oscillates around, then it’s known to diverge. The points on either side are known as the Radius of Convergence.

Important Series to Remember:

Function, $f(x)$	Expansion	Series Expansion, $\sum$	Interval of Convergence
$e^x$	${\sum }_{n=0}^{\infty }\frac{x^n}{n!}$	$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$	$(-\infty ,\infty )$
$\sin (x)$	${\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$	$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$	$(-\infty ,\infty )$
$\cos (x)$	${\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n}}{(2n)!}$	$1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$	$(-\infty ,\infty )$
$\ln (1+x)$	${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n-1}{x}^n}{n}$	$x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots$	$(-1,1)$
$\frac{1}{1-x}$	${\sum }_{n=0}^{\infty }{x}^n$	$1+x+x^2+x^3+\cdots$	$(-1,1)$
$(1+x{)}^{\alpha }$	${\sum }_{n=0}^{\infty }\frac{(\alpha {)}_n{x}^n}{n!}$	$1+\alpha x+\frac{\alpha (\alpha -1)x^2}{2!}+\frac{\alpha (\alpha -1)(\alpha -2)x^3}{3!}+\cdots$	$(-1,1)$, if $\alpha$ is not a positive integer
$\arctan (x)$	${\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{2n+1}$	$x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots$	$[-1,1]$
$\sinh (x)$	${\sum }_{n=0}^{\infty }\frac{x^{2n+1}}{(2n+1)!}$	$x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\cdots$	$(-\infty ,\infty )$
$\cosh (x)$	${\sum }_{n=0}^{\infty }\frac{x^{2n}}{(2n)!}$	$1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\cdots$	$(-\infty ,\infty )$
Bessel Function $J_0(x)$	${\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n}}{2^{2n}(n!{)}^2}$	$1-\frac{x^2}{2^2}+\frac{x^4}{2^2{4}^2}-\frac{x^6}{2^2{6}^2{4}^2}+\cdots$	$(-\infty ,\infty )$

Function	Series Expansion Notation
$\ln (1-x)$	${\sum }_{n=1}^{\infty }\frac{x^n}{n}$
$\ln (x)$	${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n-1}(x-1{)}^n}{n}$
$\ln (1+x)$	${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n-1}{x}^n}{n}$
$\ln (\frac{1+x}{1-x})$	${\sum }_{n=1}^{\infty }\frac{2x^{2n-1}}{2n-1}$
$\frac{1}{x}$	${\sum }_{n=0}^{\infty }(-1{)}^n(x-1{)}^n$
$\frac{1}{(1+x)}$	${\sum }_{n=0}^{\infty }(-1{)}^n{x}^n$
$\frac{1}{1+x^2}$	${\sum }_{n=0}^{\infty }(-1{)}^n{x}^{2n}$
$\frac{1}{1-x^2}$	${\sum }_{n=0}^{\infty }{x}^{2n}$
$\frac{1}{(1+x{)}^2}$	${\sum }_{n=1}^{\infty }(-1{)}^{n-1}n{x}^{n-1}$
$\frac{1}{(1-x{)}^2}$	${\sum }_{n=1}^{\infty }n{x}^{n-1}$
$\tan (x)$	${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n-1}{2}^{2n}(2^{2n}-1)B_{2n}{x}^{2n-1}}{(2n)!}$
$\sec (x)$	${\sum }_{n=0}^{\infty }\frac{(-1{)}^n{E}_{2n}{x}^{2n}}{(2n)!}$
$\csc (x)$	${\sum }_{n=0}^{\infty }\frac{(-1{)}^{n}2(2^{2n-1}-1)B_{2n}{x}^{2n-1}}{(2n)!}$
$\cot (x)$	${\sum }_{n=0}^{\infty }\frac{(-1{)}^n{2}^{2n}{B}_{2n}{x}^{2n-1}}{(2n)!}$