What does the notation $O(x^n)$ mean?

Question

I am reading a book about Padé Approximations, and I am trying to understand the following line:

We denote the $[L/M]$ Padé approximant to $A(x)$ by

$A(x) - P_L(x)/Q_M(x) = O(x^{L+M+1})$

where $P_L(x)$ is a polynomial of degree at most $L$ and $Q_M(x)$ is a polynomial of degree at most $M$. $A(x)$ is a formal power series $\sum\limits_{j=0}^\infty a_jx^j$

What does the $O(x^{L+M+1})$ mean?

Thanks in advance

The book is Essentials of Padé Approximants by George A. Baker, JR.

Answer 1

Usually $f(x)=O(g(x))$ means that for all sufficiently large $x$ $f(x)\leq k \cdot g(x)$. That is, $\lim \sup_{x\to\infty}\frac{f(x)}{g(x)}<\infty$.