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.
I dont think we know that there exists a sufficiently large n where those sequences are monotonically decreasing (since we don’t know that a_n is monotonically decreasing for sufficiently large n). I could be wrong here though, so if they are monotonically increasing for large enough n I’d be thankful if someone could explain it to me.