If $\sum a_n$ and $\sum \ln(1+a_n)$ converge, does $\sum a_n^2$ converge?

Let $(a_n)_{n=1}^\infty$ be a sequence with the property $a_n>-1$ for all $n$. Assume $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty \ln(1+a_n)$ converge. Must $\sum_{n=1}^\infty a_n^2$ converge also?

The Taylor polynomial of degree $2$ of $\ln(1+x)$ around $0$ is $x-\frac {x^2}{2}$, so in the opposite direction to the one requested, the convergence of both $\sum_{n=1}^\infty a_n^2$ and $\sum_{n=1}^\infty \ln(1+a_n)$ would imply convergence of $\sum_{n=1}^\infty a_n$. However I couldn't find a way to manipulate the series to get the direction in the question. I also tried the sequence $a_n= \frac{(-1)^{n+1}}{\sqrt n}$, but it doesn't seem that $\sum_{n=1}^\infty \ln(1+a_n)$ converges. How shall one continue?