Approximate the sum $S$ of the alternating series correct to within 0.0001. $$\sum\limits_{k = 0}^\infty {\frac{{{{( - 1)}^k}}}{{(2k)!}} = 1 - \frac{1}{{2!}} + \frac{1}{{4!}} - \frac{1}{{6!}} + \frac{1}{{8!}} - ...}$$

Solution We demonstrated in Example 2 that this series converges by showing that it satisfies the conditions of the Alternating Series Test. The fifth term of the series, $\dfrac{1}{8!}=\dfrac{1}{40,\!320}\approx 0.000025$, is the first term less than or equal to $0.0001$. This term represents an upper estimate to the error when the sum $S$ of the series is approximated by adding the first four terms. So, the sum $$S \approx \sum\limits_{k = 0}^3 {\frac{{{{( - 1)}^k}}}{{(2k)!}} = 1 - \frac{1}{{2!}} + \frac{1}{{4!}} - \frac{1}{{6!}} = 1 - \frac{1}{2} + \frac{1}{{24}} - \frac{1}{{720}} = \frac{{389}}{{720}} \approx 0.5403}$$

is correct to within 0.0001.