Table

Table 4: TABLE 4

Series	Number of Terms	Estimate of the Sum	Maximum Error
\[\sum\limits_{k = 1}^{^\infty } {\frac{{{{( - 1)}^{k + 1}}}}{k}} \]	\[n = 3\]	\[{S_3} = 1 - \frac{1}{2} + \frac{1}{3} \approx 0.833\]	\[\left\| {{E_3}} \right\| \le \frac{1}{4} = 0.25\]
	\[n = 9\]	\[{S_9} = \sum\limits_{k = 1}^9 {\frac{{{{( - 1)}^{k + 1}}}}{k}} \approx 0.746\]	\[\left\| {{E_9}} \right\| \le \frac{1}{{10}} = 0.1\]
	\[n = 99\]	\[{S_{99}} = \sum\limits_{k = 1}^{99} {\frac{{{{( - 1)}^{k + 1}}}}{k}} \approx 0.7\]	\[\left\| {{E_{99}}} \right\| \le \frac{1}{{100}} = 0.01\]
	\[n = 999\]	\[{S_{999}} = \sum\limits_{k = 1}^{999} {\frac{{{{( - 1)}^{k + 1}}}}{k}} \approx 0.694\]	\[\left\| {{E_{999}}} \right\| \le \frac{1}{{1000}} = 0.001\]