1. ## Help with order

Why is it possible to represent sin(x) = x + O(x^3)
thanks for any help

2. Originally Posted by hmmmm
Why is it possible to represent sin(x) = x + O(x^3)
thanks for any help
Look at the McLaurin series for sin(x): $\displaystyle \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1}= x- \frac{1}{6}x^3+ \cdot\cdot\cdot$.

3. this is where the problem arose for me because there are x^5 and so on terms in the expansion how can these all be expressed as O(x^3) or is this just an estimation??
thankyou for the help and any future help

4. Originally Posted by hmmmm
this is where the problem arose for me because there are x^5 and so on terms in the expansion how can these all be expressed as O(x^3) or is this just an estimation??
thankyou for the help and any future help

$\displaystyle \sin x = x + O(x^3) \Longleftrightarrow |\sin x - x| \leq M|x^3|\,\,\,as\,\, x \rightarrow 0\,\,\mbox{ and for some constant M}$

Using now the MacClaurin series of $\displaystyle \sin x$ that HallsofIvy wrote we get $\displaystyle \sin x -x=-\frac{x^3}{6}+\frac{x^5}{120}-...$ , and
this is less than some constant times $\displaystyle x^3$ when x is close enough to zero (can you see why?)

Tonio