Answer
$n=8$
Work Step by Step
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 an accuracy to one decimal place, we have
${a_{n + 1}} \le 0.05$
Thus,
$\dfrac{1}{{\left( {n + 2} \right)\ln \left( {n + 2} \right)}} \le 0.05$
$\left( {n + 2} \right)\ln \left( {n + 2} \right) \ge 20$
Using a calculating utility, we solve this inequality and obtain $n \ge 7.07$. Thus, the closest integer is $n=8$.
