Calc6.56

Discuss the convergence or divergence of the series $\sum _{{k=1}}^{{\infty }}a_{k}{\frac {1}{k^{3}}}$ where $a_{k}=\{7,4,6,3,-10,-10,7,4,6,3,-10,-10,...\}$.
$\{s_{n}\}=\left\{\sum _{{k=1}}^{{n}}a_{k}\right\}=\{7,11,17,20,10,0,7,11,17,20,10,0,...\}$
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.