Let $(a_n)$ be a sequence of nonegative real numbers. In this page from wikipedia as well in some question here on Math SE we see that
$$1+\sum_{n=1}^\infty a_n\leq\prod_{n=1}^\infty (1+a_n)\leq\exp\left(\sum_{n=1}^\infty a_n\right).$$
Now suppose that $\sum_{n=1}^\infty a_n$ diverges so that every term of the inequality above is $\infty$. For example for $\sum_{n=1}^\infty \frac{\lambda}{n}$ with $\lambda>1$.
My question is if there are good upper and lower bounds for $\prod_{n=1}^m (1+a_n)$, $m\in\Bbb N$.
ps.: I don't care if the bounds are ugly as hell.