Can somebody explain to me how the "Big O Notation" or the "Landau symbols" work when applied to the Taylor expansion? I'm currently reading a book about physics and I came across this—
—what does that mean? I searched the Internet but there are different explanations about that and most of them have to do with algorithms in computer science. Also the O looks different here, a little bit curved, is that the big O Notation at all? Am I correct that the O here expresses that all terms of the polynomial after that can be neglected? Or does it mean something different? Maybe someone could clarify that, I'm a bit confused :D Thanks! Jannik
Let $R(x)=f(x)-f(0)-f'(0) x$ so that $f(x)=f(0)+f'(0)x + R(x)$. What they mean by $R(x)$ having complexity $O(x^2)$ (sometimes written $R(x)=O(x^2)$) as $x \to 0$ is that for all $x$ sufficiently close to $0$, we have $|R(x)| \le C x^2$ for some constant $C$. That is, the value of $R(x)$ as $x\to 0$ is asymptotically no bigger than some multiple of $x^2$. Often times we neglect "small terms" in the Taylor expansion, but the $O(x^2)$ is there as an indication of "how small" the neglected terms are.
For example, if $R(x)=a_2 x^2 + a_3 x^3 + a_4 x^4$ then $|R(x)|/x^2 \le |a_2| + |a_3| |x| + |a_4|x^2$, and for $x$ close enough to zero, the right-hand side is smaller than $2|a_2|$. So $R(x)=O(x^2)$.