Chapter 9 - Infinite Series - 9.6 Alternating Series; Absolute And Conditional Convergence - Exercises Set 9.6 - Page 647: 34

${a_6} \approx 0.00139$

We have ${a_k} = \dfrac{1}{{k!}}$. According to Theorem 9.6.2, the absolute error $\left| {S - {s_n}} \right|$ is given by Eq. (4): $\left| {S - {s_n}} \right| \le {a_{n + 1}}$ For $n=5$, the upper bound on the absolute error is ${a_6} = \dfrac{1}{{6!}} = \dfrac{1}{{720}} \approx 0.00139$.