The series converges or diverges depending on whether $\int_2^{\infty} \frac{1}{x \ln x} dx$ converges or diverges

The series must converge because the integral $\int_2^{\infty} \frac{1}{x \ln x} dx$ diverges

The series must diverge because the integral $\int_2^{\infty} \frac{1}{x \ln x} dx$ diverges (p-series fails)

The integral test does not apply here