# Calc6.53

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.