$\int_2^{\infty} \frac{1}{x(\ln x)^2} dx = 0$ (converges)

$\int_2^{\infty} \frac{1}{x(\ln x)^2} dx = \frac{1}{\ln 2}$ (converges)

$\int_2^{\infty} \frac{1}{x(\ln x)^2} dx = \infty$ (diverges)

The Integral Test is inconclusive for this series.