Integral Test — Question 5

Question 5 – Solution

Step 1: Check the domain and hypotheses.

Start at $n=3$ so that $\ln n>1$ and $\ln(\ln n)>0$. Let $$f(x)=\frac1{x\ln x\ln(\ln x)}\quad(x\ge3).$$ The denominator is positive, continuous, and strictly increasing, so $f$ is positive, continuous, and decreasing.

Step 2: Perform two substitutions.

First let $u=\ln x$, $du=dx/x$; then let $v=\ln u$, $dv=du/u$: $$\int_3^b\frac{dx}{x\ln x\ln(\ln x)} =\int_{\ln3}^{\ln b}\frac{du}{u\ln u} =\left[\ln(\ln u)\right]_{\ln3}^{\ln b}.$$ Thus the antiderivative grows as $\ln(\ln(\ln b))$, which tends to infinity.

Step 3: Conclude.

By the Integral Test, $$\boxed{\sum_{n=3}^{\infty}\frac1{n\ln n\ln(\ln n)}\text{ diverges}.}$$ This is another borderline series: three nested logarithms still grow without bound.