calculus - does the infinite series $sum^{infty}_{n=1} (-1)^n frac {log(n)}n$ converge?

does the infinite series $\sum^{\infty}_{n=1} (-1)^n \frac {\log(n)}n$ converge?

For this one I tried absolute convergence then I applied the integral test but I realized that $\log^2(x)/2$ does not converge so I know that that won't work. Any help? Also I know the limit of $a_n$ as n approaches $\infty=0$ however I am not sure if it is non decreasing

Answer

The sequence $\log n\over n$ is decreasing for $n>2$ because the function $x\mapsto{\log x\over x}$ is decreasing in $(e,\infty)$:
$$f'(x)={1-\log x\over x^2}.$$