I need to prove or disprove the statement. I think the statement is true.
My attempt at a proof:
From the definition of convergence:
$$\forall \epsilon > 0 \quad \exists N \in \mathbb{N} \quad st \quad n,m\geq N \implies \left|\frac{a_{m+1}}{1+a_{m+1}}+\dots+\frac{a_{n}}{1+a_{n}}\right|<\epsilon$$
Since there is a finite amount of denominators and $a_n>0$ we can take the maximum:
$$g = \max\{1+a_{m+1}, \dots, 1+a_n\}$$
$$\frac{1}{g}\left|a_{m+1},\dots,a_n\right|\leq\left|\frac{a_{m+1}}{1+a_{m+1}}+\dots+\frac{a_{n}}{1+a_{n}}\right|<\epsilon$$
and that completes the proof since $\epsilon$ is arbitrary. Does this make sense?