$\lambda \cdot x = {\cal O}(x)$,
$\mu \cdot y = {\cal O}(y)$, but
$\lambda \cdot x+\mu \cdot y \neq {\cal O}(x+y)$.
I get why, but, it kind of takes the beauty out of it. I mean,
consider
$$\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}c_{kj}x^k y^j = c_{00} + c_{10}x+c_{01}y \\+ c_{11}xy<br /> +\sum_{n=2}^{\infty}\left((c_{0n}+xc_{1n})y^n+(c_{n0}+yc_{n1})x^n\right)<br /> +\sum_{k=2}^{\infty}\sum_{j=2}^{\infty}c_{kj}x^k y^j\\<br /> = c_{00} + c_{10}x+c_{01}y +{\cal O}(???)<br />$$
Is there anything elegant to write here?