Due next Thursday, 10/7, 6pm

1. Ross Ex 14.1 (briefly describe your reasoning)

2. Ross Ex 14.4

3. Let $\sum_{n=1}^\infty a_n$ be a series. Show that if $\sum_{m=1}^\infty a_{2m}$ and $\sum_{m=1}^\infty a_{2m-1}$ both converges, then $\sum_{n=1}^\infty a_n$ converges.

4. Show that if a series $\sum_n a_n$ converges absolutely, then $\sum_n a_n a_{n+1}$ converges absolutely.

5. Give an example of divergent series $\sum_n a_n$ of positive numbers $a_n$, such that $\lim_n a_{n+1} / a_n = \lim_n a_n^{1/n} = 1$. And give an example of convergent series $\sum_n b_n$ of positive numbers $a_n$, such that $\lim_n b_{n+1} / b_n = \lim_n b_n^{1/n} = 1$.