Answer
converges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{\ln n}{1.5^{n}},\ \ \ \ b_n =\frac{1}{1.4^{n}}$$
Since $\sum \frac{1}{1.4^{n}}$ is a convergent geometric series, then
\begin{align*}
\lim_{n\to \infty } \frac{a_n}{b_n } &= \lim_{n\to \infty } \frac{\ln n}{1.5^{n}} {1.4^{n}}\\
&= \lim_{n\to \infty } \ln n \lim_{n\to \infty } \frac{1.4^ n}{1.5^{n}} \\
&=0
\end{align*}
Thus, the given series also converges.
