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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.

Now, looking at the sequence of partial sums of

\{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.

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