Answer
Absolutely convergent
Work Step by Step
The Integral Test states that the series converges if and only if the integral $\int_a^\infty f(x) dx$ converges.
Let $a_n=\int_{1}^\infty \dfrac{\ln x dx}{x^3}$
Plug $a=\ln x $ and $da=\dfrac{dx}{x}$
Then $\int_{\ln 1}^{\ln \infty} \dfrac{da}{(e^{(a)})^2}=[\dfrac{e^{-2a}}{-2}]_{0}^{\infty}$
Thus $-(\dfrac{e^{-\infty}}{2})+\dfrac{e^0}{2}=0+\dfrac{1}{2}=\dfrac{1}{2}$
Thus, the series converges by the Integral Test. Also, in the given series all the terms have positive signs, and this means that the absolute series is the same as the series.
Hence, the series is absolutely convergent.
