Order of Magnitude

Defined as a non-negative function $g(h)$ for a function $f(h)$ as $h\to 0$, where ${\lim }_{h\to 0}\frac{f(h)}{g(h)}=\text{finite constant}$; also written in “big O” notation as $f(h)=O(g(h)):h\to 0$.

Order of Magnitude (Taylor series of $\cos (x)$ around $x=0$)

We define the Taylor series of cosine as $\cos (x)=1-\frac{1}{2}{x}^2+\frac{1}{24}{x}^4+\cdots +\frac{(-1{)}^n}{(2n)!}{x}^{2n}+\dots$.

If we only choose to use the first to terms of the expansion, such that $\cos (x)\approx 1-\frac{1}{2}{x}^2$, then we can determine that there is an inaccuracy of exactly the next terms: $\frac{1}{24}{x}^4+\dots$. In the limit of $x\to 0$, we can write this inaccuracy as $O(x^4)$, and hence determine the constant (in this case $\frac{1}{24}$ by using the formula definition), or simply write…

$\cos (x)=1-\frac{1}{2}{x}^2+O(x^4)$, or even $\cos (x)=1-\frac{1}{2}{x}^2+O(x^3)$ by determining that it also satisfies the equation with a constant of $0$, in this case.