Show that $\sum_{n=1}^{\infty}\ln(1+a_n)$ converges.

Question

Show that if $a_n>0$ and $\sum_{n=1}^\infty a_n$ converges, then $\sum_{n=1}^{\infty}\ln(1+a_n)$ converges too.

My attempt:

Since $a_n>0$ and $\sum_{n=1}^\infty a_n$ converges, $\lim_{n\to\infty} a_n$ must be $0$.

If we set $a_n=\ln(1+a_n)$ and $b_n = a_n$ and use the limit comparison test we have

$$
\lim_{n\to\infty} \frac{\ln(1+a_n)}{a_n}
=\lim_{x\to\infty}\frac{\ln(1+x)}{x}
=\lim_{x\to\infty}\frac{1/(1+x)}{1}=0
$$

but when the limit is $0$ we can't conclude anything about the convergence of the series, what was my mistake here?

Answer 1

Using the following property :

Under assumptions :

1. If $a_n$ is positive sequence
2. If $\sum_{\infty} a_n$ converges
3. If a sequence $b_n$ is such $a_n \sim_{\infty} b_n$

So :

$$ \sum_{\infty} b_n \ \ \text{converges} $$

Since $a_n>0$, $a_n \to 0 $ because $\sum_{\infty} a_n$ converging. $$ 0<ln(1+a_n)\sim a_n$$

You have you result.