# Calc6.58

Discuss the convergence or divergence of the series $\sum _{{k=3}}^{{\infty }}{\frac {\ln(\ln k)}{\ln k}}a_{k}$ where $\{a_{k}\}=\{1,-1,1,2,-3,1,2,4,-7,1,-1,1,2,-3,1,2,4,-7,...\}$.
It is well known that $\ln k\rightarrow \infty$ as $k\rightarrow \infty$, though it travels very slowly. So, we know that $\ln(\ln k)$ also goes to $\infty$, though much more slowly. Taking the ratio of the two, ${\frac {\ln(\ln k)}{\ln k}}$ decreases to 0 as $k\rightarrow \infty$ because the bottom increases much more quickly than the top.
$\{s_{n}\}=\left\{\sum _{{k=3}}^{{\infty }}a_{k}\right\}$ where $\{a_{k}\}=\{1,-1,1,2,-3,1,2,4,-7,1,-1,1,2,-3,1,2,4,-7,...\}$
we get the sequence of numbers $\{1,3,0,1,3,7,0,1,0,1,3,0,1,3,7,0,...\}$ (notice that k starts at 3). Since this sequence is always between 0 and 7, inclusive, it is bounded. Thus, our series converges by Dirichlet's test.