So I've been wondering the appropriate use of big-o notation (or little-o as well?) in conjunction with Taylor series.
E.g. one can write
$$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}+O(|x^6|)$$
where the $O(|x^6|)$ signifies that after this term the other terms are dominated by $|x^6|$, i.e. are smaller than $|x^6|$. I.e. if $x \rightarrow 0$ then one can figure out that the remaining terms simply vanish.
However, this raises the question. Is big-o (or little-o) useful only when the Taylor series is centered at 0? I.e. when one takes limit of $x \rightarrow 0$ or when $x \approx 0$?
No, in general if you're expanding around a point $a$, then your error terms would be related to $\mathcal{O}(|x-a|^k)$, rather than just $\mathcal{O}(|x|^k)$.