Answer
Divergent
Work Step by Step
Let us consider $f(x)=\dfrac{\ln x^2}{x}$
Here, the function $f(x)$ is positive, continuous and decreasing for $x \geq 3$
Then $\int_3^\infty \dfrac{\ln x^2}{x}dx= \lim\limits_{k \to \infty} \int_3^k \dfrac{\ln x^2}{x}dx= 2\lim\limits_{k \to \infty} [(\ln^2 (x^2))]_3^k$
and $ \lim\limits_{k \to \infty} [2(\ln^2 (k^2))-2(\ln^2 (3^2))]= \infty$
Thus, the sequence $\Sigma_{n=2}^\infty \dfrac{\ln n^2}{n}=(\dfrac{1}{2}) (\ln 4)+\Sigma_{n=3}^\infty \dfrac{\ln n^2}{n}$ is Divergent.
