Home » Blog » Prove or disprove two statements about absolute convergence

# Prove or disprove two statements about absolute convergence

Prove or disprove the following two statements.

1. If is absolutely convergent, then

is absolutely convergent.

2. If is absolutely convergent, and if for all , then,

is absolutely convergent.

1. 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

2. 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)

1. Evangelos says:

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.

• William C says:

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.