Calc6.53

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Discuss the convergence or divergence of the series \sum _{{k=7}}^{{\infty }}\left(-{\frac  {1}{2}}\right)^{k}.

We can see that this series converges since it is a geometric series with common ratio {\frac  {-1}{2}} and also because it is an alternating series with terms decreasing in magnitude. But, to give another example for Dirichlet's test, we will show this series converges by it.

\sum _{{k=7}}^{{\infty }}\left(-{\frac  {1}{2}}\right)^{k}=\sum _{{k=7}}^{{\infty }}\left(-1\right)^{k}\left({\frac  {1}{2}}\right)^{k}

This is in just the form to use Dirichlet's test. The sequence of partial sums

\{s_{n}\}=\left\{\sum _{{k=7}}^{{n}}\left(-1\right)^{k}\right\}

is bounded since it alternates between -1 and 0. Also, \left({\frac  {1}{2}}\right)^{k} decreases to 0 as k\rightarrow \infty .

Thus, by Dirichlet's test, this series converges.


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