Equivalence of convergence of a series and convergence of an infinite product 
Let $(a_n)_n$ be a sequence of non-negative real numbers.
Prove that $\sum\limits_{n=1}^{\infty}a_n$ converges if and only if the infinite product $\prod\limits_{n=1}^{\infty}(1+a_n)$ converges.

I don't necessarily need a full solution, just a clue on how to proceed would be appreciated.
Could any convergence test like the ratio or the root test help?

A way is to use logarithm and the equivalence
$$\ln(1+x) \sim x$$ around the origin.
Interesting to notice is that for complex series (or series not having a constant sign) $\sum z_n$, the absolute convergence of the series implies the convergence of the product $\displaystyle \prod (1+z_n)$, but the converse is not always true. See my counterexamples web site.
