Prove or disprove two statements about absolute convergence

by RoRi

March 18, 2016

Prove or disprove the following two statements.

If $\sum a_n$ is absolutely convergent, then
$\sum \frac{a_n^2}{(1+a_n^2)}$

is absolutely convergent.
If $\sum a_n$ is absolutely convergent, and if $a_n \neq -1$ for all $n$ , then,
$\sum \frac{a_n}{1+a_n}$

is absolutely convergent.

Proof. Since $\sum a_n$ converges absolutely, we know $\sum |a_n|$ converges. By a previous exercise (Section 10.20, Exercise #50) we know that this implies that $\sum a_n^2$ converges. Then we have,
$\frac{a_n^2}{1+a_n^2} \leq a_n^2$

for all $n$ since $1+ a_n^2 \geq 1$ . Hence, by comparison with $\sum a_n^2$ we have the convergence of

$\sum_{n=1}^{\infty} \frac{a_n^2}{1+a_n^2}. \qquad \blacksquare$
Proof. Since $\sum a_n$ converges absolutely, we know $\sum |a_n|$ converges. Then, using the limit comparison test we have,
$\begin{align*} \lim_{n \to \infty} \frac{|a_n|}{\left| \frac{a_n}{1+a_n} \right|} &= \lim_{n \to \infty} |1 + a_n| \\[9pt] &= 1. \end{align*}$

The final equality follows since $\sum a_n$ converges means we must have $\lim_{n \to \infty} a_n = 0$ . Hence, the series $\sum \left| \frac{a_n}{1+a_n} \right|$ converges; and thus,

$\sum \frac{a_n}{1+a_n}$

converges absolutely (as long as $a_n \neq -1$ so that all of the terms are defined) $. \qquad \blacksquare$

Related

2 comments

May 14, 2021 at 9:13 am

Evangelos says:

We can also use Abel’s test for these, given that n is sufficiently large such that

$|\frac{1}{1 + a_{n}}|$

and

$\frac{1}{1 + a_{n}^{2}}$

is monotonic decreasing, with limit 1 in both cases.

Reply
- March 14, 2023 at 9:09 am
  
  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.
  
  Reply

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): Cancel reply