Background:
In the text book I have, in the 'expansion of integrands' part (perturbation theory) the author kept determining the order, but I can't understand how.
For integration of $\sin \epsilon x^2$, the Taylor series is written, $$<first\ three\ terms> +\ O(\epsilon^7)$$ (I haven't gone through the tedious task of typing the first three terms.) Then it is written in the book,
...Since $|x|\le1$ and $\epsilon$ is small, the remainder term in the above series is $O(\epsilon^7)$ for all values of $x%$.
In every example, something like this is written. I can't understand the flow of logic here.
I have searched the internet to understand Taylor series and remainder term better, but all I get is theorem regarding this. I want to know how explicitly it is determined just when the series needs to be truncated. I want to know 'when' to truncate, more than 'how' to truncate.
Question:
- In the above example, why is the series truncated after $O(\epsilon^5)$? Why not after $O(\epsilon^3)$ or $O(\epsilon^7)$?
- What is the relation between $\epsilon$ being small and the order of the remainder term being $O(\epsilon^7)$? If I truncated, say after $O(\epsilon^{3})$, $\epsilon$ was also then small, and then the remainder term would have been $O(\epsilon^{5})$.
- How to find when to truncate a series for any function? I thought I could truncate any series acording to my need and wish. Seems like, at least in expansion of integrands, I am wrong, there is a logic for that.