University Calculus: Early Transcendentals (3rd Edition)

$\Sigma b_n$is a convergent series
We are given that $\Sigma a_n$ is convergent. and $\Sigma a_n \gt 0$ and $\Sigma b_n \gt 0$ Since, we have $\lim\limits_{x \to \infty} \dfrac{a_n}{b_n} =\infty$ Thus, the series $\Sigma b_n$ must also be a convergent series by the limit comparison test. Hence, the given series $\Sigma b_n$is convergent.