Chapter 5.5, Problem 305E

The following series do not satisfy the hypotheses of the alternating series test as stated.

In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.

305. Sometimes the alternating series ${\displaystyle \underset{n=}{\overset{\infty }{\sum }}{\left(-1\right)}^{n-1}{b}_n}$ converges to a certain fraction of an absolutely convergent series ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{b}_n}$at a faster rate. Given that ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n^2}-\frac{{\pi }^2}{6}}$ , find $S=1-\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\mathrm{...}$ Which of the series $6{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n^2}}$and $S{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^{n-1}}{n^2}}$ gives a better estimation

of ${\pi }^2$ using 1000 terms?