Calc6.54

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Discuss the convergence or divergence of the series \sum _{{k=1}}^{{\infty }}{\frac  {1}{k2^{k}}}.

The sequence of partial sums

\{s_{n}\}=\left\{\sum _{{k=1}}^{{n}}{\frac  {1}{2^{k}}}\right\}

converges to 1 since it is essentially a geometric series. Thus, it is bounded.

Since {\frac  {1}{k}} decreases to 0 as k\rightarrow \infty , this series converges by Dirichlet's test.


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