Maclaurin Series — Last Minute Summary

Formula

$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$

Standard Expansions (must memorise)

Function	Expansion (first terms)
$e^x$	$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$
$\sin x$	$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$
$\cos x$	$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$
$\ln(1+x)$	$x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$
$\tan^{-1}x$	$x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots$
$\sinh x$	$x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots$
$\cosh x$	$1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots$

Key Tricks

• $a^x = e^{x\ln a}$
• For composite $f(g(x))$, expand $f(u)$ then substitute $u=g(x)$
• For $\ln(\cosh x)$, expand $\cosh x$ then use $\ln(1+u)$ series
• For $\sin^{-1}x$, differentiate first then integrate series

Derivatives to Remember

$\frac{d}{dx}a^x = a^x \ln a$

$\frac{d}{dx}\sin^{-1}x = \frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}\tan^{-1}x = \frac{1}{1+x^2}$

$\frac{d}{dx}\ln(\cosh x) = \tanh x$

Warning Signs

Expression	Trap
$a^x$	$\neq xa^{x-1}$, correct: $a^x\ln a$
$\sin^{-1}x$	derivative has $1/\sqrt{1-x^2}$
$\ln(\cosh x)$	need $\cosh x$ expansion first
$e^{1+x^2}$	factor $e$ first: $e \cdot e^{x^2}$