Is the ideal $(X^2-3)$ proper in $\mathbb{F}[[X]]$? 
Let $\mathbb{F}$ be a field and $R=\mathbb{F}[[X]]$ be the ring of formal power series over $\mathbb{F}$. Is the ideal $(X^2-3)$ proper in $R$? Does the answer depend upon $\mathbb{F}$?

Clearly $X^2-3=(X+\sqrt3)(X-\sqrt3)$ and hence $X^2-3$ is not zero.
I have no idea whether the ideal is proper or not. 
So far, I didn't learn any theorem to prove an ideal is proper. Perhaps I should start with definition of proper ideal? Find one element in $R$ but not in the ideal ?

HINT: $$\frac1{x^2-3}=-\frac13\cdot\frac1{1-\frac{x^2}3}=-\frac13\sum_{x\ge 0}\frac{x^{2n}}{3^n}\in R\text{ provided that}\dots ?$$
