Chapter 5.5, Problem 309E

The following alternating series converge to given multiples of $\pi$ . Find the value of N predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum N for which the error bound holds, and give the desired approximate value in

each case. Up to 15 decimals places,

$\pi$ =3.141592653589793…

309. $\left[\text{T}\right]\text{if}{\displaystyle \underset{n=1}{\overset{N}{\sum }}{\left(-1\right)}^{n-1}\frac{1}{n}\to \ln 2,}$what is $1+\frac{1}{3}+\frac{1}{5}-\frac{1}{2}-\frac{1}{4}-\frac{1}{6}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\frac{1}{8}-\frac{1}{10}-\frac{1}{12}+\mathrm{...}?$