Let $P(x)=1+a_1x+a_2x^2+a_3x^3+...$ be a series such that every $a_i$ is an integer, $a_1<0$, and $a_i\ge 0$ for every $i\ge 2$. Are the following statements equivalent ?

- $P(y)=0$ for some $y>0$.
- Every coefficient of the series expansion of $\frac 1{P(x)}$ is positive.