Infinite Product is converges

Question

I am adding this problem since it is interesting and valuable to be verified here:

Prove that the infinite product $\prod_{k=1}^{\infty}(1+u_k)$, wherein $u_k>0$, converges if $\sum_{k=1}^{\infty} u_k$ converges. What about the inverse problem?

Thanks for any ideas.

Answer 1

Note that since $u_k > 0$, we have $$\sum_{k=1}^n u_k \leq \prod_{k=1}^n (1+u_k) \leq \exp \left(\sum_{k=1}^n u_k \right)$$ Hence, if $u_k>0$, we have that$\displaystyle \prod_{k=1}^\infty (1+u_k)$ converges iff $\displaystyle \sum_{k=1}^\infty u_k$ converges.