IB Maths AA 5.13 Notes

Maclaurin formula

A Maclaurin series writes a function as a power series centred at $x=0$. In this topic, the required standard expansions are for $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$ and $(1+x)^p$, where $p\in\mathbb{Q}$.

$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+\cdots$

More compactly,

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

This gives a polynomial-style expansion that approximates the function near $x=0$.

Some standard expansions include:

• $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots$
• $\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$
• $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$
• $\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots$, valid for $-1\lt x\le1$
• $(1+x)^p=1+px+\frac{p(p-1)}{2!}x^2+\frac{p(p-1)(p-2)}{3!}x^3+\cdots$

If $p$ is not a positive integer, the binomial-type expansion is valid for $|x|\lt 1$. If $p$ is a positive integer, the series stops and becomes an exact polynomial.