Chapter 9 - Infinite Series - 9.6 Alternating Series; Absolute And Conditional Convergence - Exercises Set 9.6 - Page 647: 36

${a_4} \approx 0.124$

We have ${a_k} = \dfrac{1}{{\left( {k + 1} \right)\ln \left( {k + 1} \right)}}$. 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 $n=3$, the upper bound on the absolute error is ${a_4} = \dfrac{1}{{5\ln 5}} \approx 0.124$.