If $\sum_\limits{n\in\mathbb{N}}{b_n}$ is divergent, then so is $\sum_\limits{n\in\mathbb{N}}{\dfrac{b_n}{1+b_n}}$?
Suppose $\sum\limits_{n=1}^\infty b_n$ diverges and $b_n\gt 0$, show that the series $\sum\limits_{n=1}^\infty \frac{b_n}{1+b_n}$ also diverges