Calc6.56

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Discuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}a_{k}{\frac  {1}{k^{3}}} where a_{k}=\{7,4,6,3,-10,-10,7,4,6,3,-10,-10,...\}.

This series is not like any series you normally see in calculus books. However, it is easily shown to converge by Dirichlet's test. First, we have the sequence of partial sums

\{s_{n}\}=\left\{\sum _{{k=1}}^{{n}}a_{k}\right\}=\{7,11,17,20,10,0,7,11,17,20,10,0,...\}

Thus, this sequence ranges from 0 to 20 but never gets outside of this range. That is, it is bounded. Further, {\frac  {1}{k^{3}}} decreases to 0 as k\rightarrow \infty . Thus, this series converges by Dirichlet's test.


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