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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.

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See also this blog post: Convergence of Infinite Products by Jim Belk or this answer by Marvis. – Martin Sleziak Nov 24 '12 at 8:08
@MartinSleziak: Thank you for noting me that. – Babak S. Nov 24 '12 at 8:12
up vote 2 down vote accepted

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.

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Sorry for asking, but are you using $1+x\leq \text{exp}(x)$ for $x>0$. And what about the inverse direction? Thanks Marvis. – Babak S. Nov 24 '12 at 8:05
@BabakSorouh Yes. I am using the fact that $1+x \leq \exp(x)$. What do you mean by the inverse direction? – user17762 Nov 24 '12 at 8:06
It is done. I see "iff" above in the answer. :) – Babak S. Nov 24 '12 at 8:08

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