Convergence of Improper Integral: $\int_{e^2}^\infty {dx\over x\log\log x}$ Test the convergence of the following integral$$\int_{e^2}^\infty {dx\over x\log\log x}$$
I understand that the problem is only at $\infty$ how to proceed ?

The integral converges iff the series
$$\sum_{n>e^2}\frac1{n\log\log n}\;\;\text{converges}$$
Applying the condensation test to this series, we get it converges iff
$$\sum_{n>e^2}\frac{2^n}{2^n\log\log2^n}=\sum_{n>e^2}\frac1{\log n+\log\log2}$$
and this last series diverges...
