Chapter 1.7, Problem 10P

The following alternating series are divergent (but you are not asked to prove this). Show that an $a_n\to 0$.

Why doesn’t the alternating series test prove (incorrectly) that these series converge?

(a). $2-\frac{1}{2}+\frac{2}{3}-\frac{1}{4}+\frac{2}{5}-\frac{1}{6}+\frac{2}{7}-\frac{1}{8}\mathrm{...}$

(b). $\frac{1}{\sqrt{2}}-\frac{1}{2}+\frac{1}{\sqrt{3}}-\frac{1}{3}+\frac{1}{\sqrt{4}}-\frac{1}{4}+\frac{1}{\sqrt{5}}-\frac{1}{5}\mathrm{...}$