The following properties of Landau notation hold:

1. $o(f)$ and $O(f)$ are vector spaces, i.e. if $g,h\in o(f)$ (resp. in $O(f)$ ) then $\lambda g + \mu h \in o(f)$ (resp. in $O(f)$ ) whenever $\lambda,\mu \in \R$ ; In particular $o(f) + o(f) = o(f)$ and $\lambda o(f) = o(f)$ ;
2. if $\lambda\neq 0$ then $\lambda o(f) = o(f)$ and $\lambda O(f)=O(f)$ ;
3. $f o(g)=o(f g)$ , $f O(g)= O(f g)$ ;
4. $o(g)^\alpha = o(g^\alpha)$ , $O(g)^\alpha = O(g^\alpha)$ ;
5. $o(f) \subseteq O(f)$ ; on the other hand if $f\in o(g)$ then $O(f)\subseteq o(g)$ ;
6. $o(f)\subseteq o(g)$ if $f\in O(g)$ ; analogously $O(f)\subseteq O(g)$ if $f\in O(g)$ ;
7. $o(o(f))=o(f)$ , $O(O(f))=O(f)$ , $o(O(f))=o(f)$ , $O(o(f))=o(f)$ .

Here are some examples. First of all we consider Taylor formula. If $x_0\in (a,b)\subset \R$ and $f\colon (a,b)\to \R$ has $n$ derivatives, then $$ f(x) \in \sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!} (x-x_0)^k + o((x-x_0)^n). $$ As a consequence, if $f$ has $n+1$ derivatives, we can replace $o((x-x_0)^n)$ with $O((x-x_0)^{n+1})$ in the previous formula.

For example: $$ e^x \in 1 + x + \frac 12 x^2 + \frac 16 x^4 + O(x^5) \subset 1 + x + \frac 12 x^2 + \frac 16 x^4 + o(x^4). $$

Using the properties stated above we can compose and iterate Taylor expansions. For example from the expansions $$ \sin x \in x + \frac {x^3}{3!} + o(x^4),\qquad e^x \in 1 + x + \frac{x^2}{2} + O (x^3), $$ $$ \cos x \in 1 - \frac{x^2}{2} + \frac {x^4}{4!} + o (x^5) \subseteq 1- \frac{x^2}{2} + O(x^4),\qquad \log( 1+x) \in x - \frac{x^2}{2} + o (x^2) $$ we get