The series diverges because $\int_2^{\infty} \frac{\ln x}{x^2} dx$ diverges

The series converges because $\int_2^{\infty} \frac{\ln x}{x^2} dx$ converges to a finite value

The integral diverges, but the series converges by the Limit Comparison Test

The Integral Test does not apply because $\ln n$ is not eventually decreasing