Calc6.51

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Discuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}{\frac  {\sin {\frac  {k\pi }{2}}}{k}}.

To show convergence for this series we will use Dirichlet’s test. The terms of this sequence are the product of {\frac  {1}{k}} and \sin {\frac  {k\pi }{2}}. The first sequence of numbers decreases toward 0 as k\rightarrow \infty . For the second sequence of numbers, \sin {\frac  {k\pi }{2}}, let us look at the partial sums

s_{n}=\sum _{{k=1}}^{{n}}\sin {\frac  {k\pi }{2}}\,

The terms of the sequence are {1, 1, 0, 0, 1, 1, 0, 0, ...}. Thus, this second sequence of numbers has bounded partial sums. Since our series has terms which are a product of the terms of one sequence which decreases to 0 and another which has bounded partial sums, by Dirichlet’s test, this series converges.


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