What is O in binomial distribution entropy?

Question

I have searched everywhere but can't find an answer. What is "O" refers to in this equation?

$${\frac {1}{2}}\log _{2}\left(2\pi enp(1-p)\right)+O\left({\frac {1}{n}}\right)$$

Answer 1

It is the "big Oh" notation described in this wikipedia article. In your case it means an approximation error of size no worse then $\pm C/n$, for some unstated constant $C$.

More precisely, such an estimate holds for all sufficiently large $n$. Your example is derived from Stirling's approximation to $\log n!$; this approximation has only limited accuracy for small $n$ but better and better accuracy for big $n$.

Answer 2

Let $f(n) = O\left(\frac 1 n \right)$ then $|f(n)| \le \frac M n$ for all $n \ge n_0$, for some positive constant $M$ and $n_0$.