Let $p$ be a polynomial with no constant term. If $b\_j > 0$ for every $j$ and if $\displaystyle\sum\_{j=1}^{\infty}{b\_j}$ converges then prove that the series $\displaystyle\sum\_{j=1}^{\infty}{p(b\_j)}$ converges.

I immediately ruled out ratio test as the approach since I can do nothing with polynomials in both numerator and denominator. I guess root test is the right path since each term is a polynomial and we can just pick out some terms after expansion. But the degree of each term being different is troublesome, and I fail to find a way to develop inequalities that connect with the condition given.

Ideas?