Prove or disprove: $a_n >0$, $\sum _{n=1}^\infty \ln(1+a_n)$ is convergent $\Rightarrow$ $\sum _{n=1}^\infty a_n$ is convergent.

We know that $\sum _{n=1}^\infty \ln(1+a_n)$ is convergent if $\sum _{n=1}^\infty a_n$ is convergent where $a_n >0$ for all $n\geq 1$. This follows by using the inequality $\ln(1+x) \leq x$ for all $x\geq 0$ and the comparison test. I am trying to see if the converse is true, i.e., whether the convergence of $\sum _{n=1}^\infty \ln(1+a_n)$ implies the convergence of $\sum _{n=1}^\infty a_n$.

We have $$\lim_{x \to 0} \frac{\log(1+x)}{x} = 1$$ This implies, under the assumption $a_n>0$, that $$\sum_{n=1}^\infty \log(1+a_n)$$ converges if and only if $$\sum_{n=1}^\infty a_n$$ converges, by the Limit comparison test.

You can use the estimate $$\exp( \log (1+a_1) + \cdots + \log(1+a_n))=(1+a_1) \cdots (1+a_n) \ge 1 + (a_1 + \cdots + a_n)$$