If $\sum_{n=1}^\infty \ln(1+a_n)$ coverges, does $\sum_{n=1}^\infty a_n$ also converge?

Question

If $\sum_{n=1}^\infty \ln(1+a_n)$ coverges and $a_n>0$ for all $n>0$, does $\sum_{n=1}^\infty a_n$ also converge? I know that the converse of this is true, but would this also be true?

Answer 1

In spirit, since $\sum_n\ln(1+a_n)$ converges, you have that $a_n\to0$, and then $\ln(1+a_n)\sim a_n$.

Formally, one can show that if $0<x<1$, then $\ln(1+x)\geq x/2$. So, for $k $ big enough, $$ \sum_{n\geq k}a_n\leq2\,\sum_{n\geq k}\ln(1+a_n)<\infty. $$

For a proof of the inequality $\ln(1+x)\geq x/2$, note that they agree at $x=0$, and the derivative of the left-hand-side is greater than the derivative of the right-hand-side. Or, one can use the Mean Value Theorem: since $0<x<1$ there exists $c(x)\in(0,1)$ (more properly, $0<c(x)<x$) with $$ \ln(1+x)=\ln(1+x)-\ln 1=\frac1{1+c(x)}\,(x-0)\geq\frac x2. $$