Q:

Polynomial of the term of convergent series converges

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?

A:

Since $\sum_{j=1}^{\infty}{b_j}$ converges, we have 
$\lim_{j\to\infty}b_j=0.$
In particularly, $b_j$ is bounded, i.e. $$\tag{1}|b_j|\leq M.$$
By assumption, $p$ is a polynomial with no constant term, i.e. 
$$p(x)=a_n x^n+\cdots a_2 x^2+a_1 x,$$
which implies
$$|p(b_j)|=\Big|(a_n b_j^{n-1}+\cdots a_2 b_j+a_1) b_j\Big|\leq 
\Big(|a_n| M^{n-1}+\cdots |a_2| M+|a_1|\Big)b_j$$
by $(1)$ and the assumption that $b_j>0$. 
Since $\sum_{j=1}^{\infty}{b_j}$ converges,  $\sum_{j}|p(b_j)|$ converges by comparison test. And since $\sum_{j}|p(b_j)|$ converges, $\sum_{j}p(b_j)$ converges. 

