Determine the convergence of the series ∑n=1∞an\sum_{n=1}^{\infty} a_n∑n=1∞an where an=∫nn+11xlnxln(lnx)dxa_n = \int_{n}^{n+1} \frac{1}{x \ln x \ln(\ln x)} dxan=∫nn+1xlnxln(lnx)1dx.
Converges by the Integral Test
Diverges because the integrand is monotonically decreasing
Converges by the Comparison Test with 1/n21/n^21/n2
Diverges because ∫e∞1xlnxln(lnx)dx\int_e^{\infty} \frac{1}{x \ln x \ln(\ln x)} dx∫e∞xlnxln(lnx)1dx diverges