Apply the Integral Test to ∑n=3∞1nlnnlnlnn\sum_{n=3}^{\infty} \frac{1}{n \ln n \ln \ln n}∑n=3∞nlnnlnlnn1.
Evaluate ∫3∞dxxlnxlnlnx\int_3^{\infty} \frac{dx}{x \ln x \ln \ln x}∫3∞xlnxlnlnxdx and determine convergence.
The integral equals ln(lnlnx)∣3∞\ln(\ln \ln x)|_3^{\infty}ln(lnlnx)∣3∞, which diverges, so the series diverges.
The integral equals lnlnx∣3∞\ln \ln x|_3^{\infty}lnlnx∣3∞, which converges, so the series converges.
The integral equals (lnlnx)2∣3∞(\ln \ln x)^2|_3^{\infty}(lnlnx)2∣3∞, which diverges, so the series diverges.
The integral equals a finite value, so the series converges.