Let $a_k \geq 0$ be a sequence, then show that $\sum_{i = 0}^\infty a_i < \infty$ implies $\prod_{i = 0}^\infty (1 + a_i) < \infty$.

Asked Oct 08 '20 at 16:16

Active Oct 08 '20 at 16:16

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I am trying to prove a claim:

Let $a_i \geq 0$ be a sequence, then $\sum_{i = 0}^\infty a_i < \infty$ implies $\prod_{i = 0}^\infty (1 + a_i) < \infty$

We can proceed by writing out the first few products...

$(1+a_0)(1+a_1) = (1 + a_0 + a_1 + a_0a_1)$

$(1+a_0)(1+a_1)(1+a_2) = ((1 + a_0 + a_1) + a_0a_1)(1+a_2) =(1 + a_0 + a_1)(1+a_2) + a_0a_1 + a_0a_1a_2 = 1+a_0+a_1+a_2 + a_0a_2 + a_1a_2 + a_0 a_1 + a_0a_1a_2$

...

So in general, it seems that $\prod_{i = 0}^\infty (1+a_i) = 1 + \sum_{i = 0}^\infty a_i + \text{cross terms} + \prod_{i = 0}^\infty a_i$. However $\prod_{i = 0}^\infty a_i$ blows up so I cannot see how it is bounded.

Does anyone see how to proceed from here?

asked Oct 08 '20 at 16:16

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Note that (if $a_i\geq 0$) $\prod_{i}(1+a_i)<+\infty$ iff $\sum_{i}\ln(1+a_i)<+\infty$. What can you say about $\sum_{i}\ln(1+a_i)$ and $\sum_{i}a_i$? – richrow Oct 08 '20 at 16:19
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Hint: $1+a \le e^a$ – achille hui Oct 08 '20 at 16:19
Hint: What standard function defined on positive numbers do you recall that takes multiplication to addition? And what properties do you recall of it? – Jack LeGrüß Oct 08 '20 at 16:21
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$\prod_{i = 0}^\infty a_i$ doesn't blow up; it is zero. But that's irrelevant here, because @achillehui's hint tells you all you need to know. – TonyK Oct 08 '20 at 16:22
See also Wikipedia. – metamorphy Oct 08 '20 at 16:31
This has been answers here in this site. – Mittens Oct 08 '20 at 16:48

Let $a_k \geq 0$ be a sequence, then show that $\sum_{i = 0}^\infty a_i < \infty$ implies $\prod_{i = 0}^\infty (1 + a_i) < \infty$.

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