Prove or disprove the following two statements.
- If is absolutely convergent, then
is absolutely convergent.
- If is absolutely convergent, and if for all , then,
is absolutely convergent.
- Proof. Since converges absolutely, we know converges. By a previous exercise (Section 10.20, Exercise #50) we know that this implies that converges. Then we have,
for all since . Hence, by comparison with we have the convergence of
- Proof. Since converges absolutely, we know converges. Then, using the limit comparison test we have,
The final equality follows since converges means we must have . Hence, the series converges; and thus,
converges absolutely (as long as so that all of the terms are defined)
We can also use Abel’s test for these, given that n is sufficiently large such that
and
is monotonic decreasing, with limit 1 in both cases.