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: $$\sum_{n\geq1}\frac{a_n}{1+a_n}=+\infty.$$

We can divide into  cases:


*

*If a(n) has limit zero : It is lower than 1 for all n bigger than n0, then we can compare with a(n)/2 which is lower than  a(n)/(1+a(n)).

*If a(n) has limit different to zero , also a(n)/1+a(n) and then the series diverges

*If a(n) is not bounded it ha a subsequence that converges to infinite,
then a(n)/1+a(n) converges to 1 then the series diverges to infinite.

*If a(n) is bounded , we can take a subsequence that is convergent.
If it does not converges to zero also the sequence a(n)/1+a(n).
If all subsequences converge to zero ,then also a(n) and we can apply 1.
