Outcome • Remember the general Taylor formula and the Maclaurin special case • Know how to grab the first n terms for quick approximations • Recall the remainder term’s shape to bound your error
Definition
For a function $f(x)$ with enough derivatives near $a$:
$$ f(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{k!}(x-a)^{k} $$ • If $a=0$ it’s called a Maclaurin series.
The $n^{\text{th}}$-degree Taylor polynomial:
$$ P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^{k} $$
Remainder / error after $n$ terms (Lagrange form):
$$ R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{,n+1}, \quad \xi \text{ between } a \text{ and } x $$
Classic examples
Function First few Maclaurin terms $\displaystyle e^{x}$ $1+x+\dfrac{x^{2}}{2!}+\dfrac{x^{3}}{3!}+\cdots$ $\displaystyle \sin x$ $x-\dfrac{x^{3}}{3!}+\dfrac{x^{5}}{5!}-\cdots$ $\displaystyle \cos x$ $1-\dfrac{x^{2}}{2!}+\dfrac{x^{4}}{4!}-\cdots$
Quick hack: drop after $x^{3}$ for $\sin x$ when $|x|\le 0.2$ — error $<10^{-4}$ thanks to $R_3$.
How to grind it fast 1. Take derivatives until the pattern is obvious. 2. Evaluate at $a$ (often $0$). 3. Plug into the coefficient formula $\dfrac{f^{(k)}(a)}{k!}$. 4. Stop at the term your exam asks for; quote the remainder to justify accuracy.
Mnemonic: “Derive, Plug, Divide, Write.”
Micro-practice
Approximate $\ln(1+x)$ near $x=0$ up to $x^{3}$:
$$ f(x)=\ln(1+x), ; f’(x)=\frac1{1+x},;f’’(x)=-\frac1{(1+x)^2},;f’’’(x)=\frac{2}{(1+x)^3} $$
At $x=0$: $f(0)=0,;f’(0)=1,;f’’(0)=-1,;f’’’(0)=2$.
So
$$ \ln(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}+R_3(x) $$
with
$$ |R_3(x)|\le\frac{2|x|^{4}}{4!(1+\xi)^{4}}\le\frac{|x|^{4}}{12} $$
(for $|x|\le 0.5$, error $<0.002$).
Source: lecture PDF “Taylor Series”