positive series convergent properties So I have a test in the next weak and I found myself struggled in an "easy" question.
Given two series $(a_n)_{n=1}^{\infty} \ $ so that $\forall n \in \mathbb{N}  \ a_n > 0$ then:
$ \sum _{n=1}^{ \infty} a_n$ convergent iff
$ \sum _{n=1}^{ \infty} \frac{a_n}{a_n +1}$ convergent.
So I know that $ \forall n \in \mathbb{N} \ \ \frac{\frac{a_n}{a_n + 1}}{\frac{a_n}{1}} = \frac{1}{a_n +1} < \frac{1}{1} = 1$
Assuming $ \sum _{n=1}^{ \infty} a_n$ convergent
then I know that $ \ a_n \rightarrow 0$
and therefore $ \ \frac{1}{a_n +1} \rightarrow 1 \ $ and because we know that $ \sum _{n=1}^{ \infty} a_n$ convergent we conclude from the 
comparison test that $ \sum _{n=1}^{ \infty} \frac{a_n}{a_n +1}$ convergent.
But what about the opposite ?
What can I conclude from the fact that $ \ \frac{a_n}{a_n +1} \rightarrow 0 \ $
That could help me proof that $ \ \frac{1}{a_n +1} \rightarrow L_1 > 0 \ $
So that $ \sum _{n=1}^{ \infty} a_n$ convergent ?
If I have to proof the other way in a different way' please tell me how .
Thanks in advanced !!

Hint: $$\frac{1}{a_n+1} = 1 - \frac{a_n}{a_n+1}.$$
