Positive series problem: $\sum\limits_{n\geq1}a_n=+\infty$ implies $\sum_{n\geq1}\frac{a_n}{1+a_n}=+\infty$ Let $\sum\limits_{n\geq1}a_n$ be a positive series, and $\sum\limits_{n\geq1}a_n=+\infty$, prove that: $$\sum_{n\geq1}\frac{a_n}{1+a_n}=+\infty.$$

Suppose $\sum{a_n\over 1+{a_n}}$ converges. Then $\frac{a_n}{1+a_n}\to 0$.
It is easy to see that:
$$\lim_{n\to\infty}a_n=0\iff \lim_{n\to\infty}\frac{a_n}{1+a_n}=0.$$
(let $b_n=\frac{a_n}{1+a_n}$, then $a_n=\frac{b_n}{1-b_n}$
)
So we have
$$\lim_{n\to\infty}\frac{\frac{a_n}{1+a_n}}{a_n}=1,$$
by comparision test of positive series,
the series $\sum a_n$ converges, it is a contradiction.
