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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.

Martin Sleziak
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Mikasa
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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.

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