Calc6.52

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Discuss the convergence or divergence of the series \sum _{{k=2}}^{{\infty }}{\frac  {\sin {\frac  {k\pi }{3}}}{\ln k}}.

Here we have {\frac  {1}{\ln k}} which goes to 0 as k\rightarrow \infty and \sin {\frac  {k\pi }{3}} for which the sequence of partial sums is bounded.

\{s_{n}\}=\{\sum _{{k=2}}^{{n}}\sin {\frac  {k\pi }{3}}\}=\{{\frac  {{\sqrt  {3}}}{2}},{\frac  {{\sqrt  {3}}}{2}},0,-{\frac  {{\sqrt  {3}}}{2}},-{\frac  {{\sqrt  {3}}}{2}},0,{\frac  {{\sqrt  {3}}}{2}},{\frac  {{\sqrt  {3}}}{2}},...\}\,

Thus, by Dirichlet’s test this series converges.


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