Another simple series convergence question: $\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$

Question

I'm being asked to determine if $\displaystyle\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$ converges. So, using Cauchy's Condensation Test, I reduced the problem to one of determining the convergence of $\displaystyle\sum\limits_{n=3}^\infty\frac 1{n\ln (n\ln 2)}$. Am I on the right path, and how do I proceed from here?

Answer 1

If $a_n$ is eventually nonincreasing, then $\sum a_n$ converges iff $\sum 2^na_{2^n}$ converges (Cauchy condensation test). This takes us from $$\sum\frac 1{n\ln n \ln\ln n}$$ to $$\sum\frac {2^n}{2^n\cdot n\ln 2 \cdot (\ln n + \ln\ln 2)}=\sum\frac {1}{n\ln 2 \cdot (\ln n + \ln\ln 2)}$$ and then to $$\sum\frac {2^n}{2^n\ln 2 \cdot (n \ln 2 + \ln\ln 2)}=\sum\frac {1}{n\ln^2 2 + \ln2\cdot\ln\ln 2}$$ which is essentially the harmonic series.

Answer 2

Since $\ln 2 $ is constant $\frac 1{n\ln (n\ln 2)} $ is comparable to $\frac 1{n\ln (n)}$ you can use integral test here.

Answer 3

Observe that

$$\frac{d}{dx}(\ln\ln x)=\frac1{\ln x}\cdot\frac{d}{dx}(\ln x)=\frac1{x\ln x}$$

and use the integral test.

Answer 4

Apply directly integral test

$$\int_3^{\infty}\frac{(\ln(\ln x))'}{ \ln(\ln x)}\mathrm{dx}= \left[\ln(\ln(\ln(x)))\right]_3^{\infty}\longrightarrow \infty$$