Chapter 5.5, Problem 280E

In each of the following problems, use the estimate $\left|R_N\right|\le b_{N+1}$to find a value of N that guarantees that the sum of the first N terms of the alternating series

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{\left(-1\right)}^{n+1}{b}_n}$differs from the infinite sum b at most the given error. Calculate the partial sum $S_N$ for this N.

280. $\left[\text{T}\right]b_n=1/n,\text{error}{\text{<10}}^{\text{-5}}$