What's the meaning of ($x^2$) when we compute a Taylor's polynomial?

Question

I'm confused. I've seen people write "+($x^2$)" at the end of a Taylor Expansion countless times and I understand it has to do with the uncalculated terms, but where does the $x^2$ come from? How does it work?

Answer 1

${\cal O}(x^2)$ simply means terms that are of order $x^2$ or higher. This is called "big-oh" notation, which has a formal definition. Anyway, informally:

Thus $f(x) = a + b x + c x^2 + d x^3 + e x^4$ can be written $f(x) = a + b x + {\cal O}(x^2)$.

You could also write $f(x) = a + b x + c x^2 + {\cal O}(x^3)$

... and so on.