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How can I prove that the infinite product

$$\displaystyle\prod_{n=1}^{+\infty}(1+z^{2n})$$

diverges for $|z|>1$?

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2 Answers 2

up vote 4 down vote accepted

$|1+z^{2n}|\geqslant |z|^{2n}-1\to +\infty$, so the partial products cannot converge.

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Irrelevant to your question but, its to big to put in a comment,

I thought I should mention the nice identity $$\prod_{k=0}^\infty(1+x^{2^k})=\frac{1}{1-x}\text{ , for all x < 1}$$

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yes, i just proved this identity, but then i was asked to prove the divergence and since it looked too obvious i thought i was missing something, so i asked –  bateman Jan 5 '13 at 10:19
    
@Ethan: As far as I know, you are not allowed to publicly ask about errors. Of course, if you were to post the question anonymously... –  Zev Chonoles Nov 2 '13 at 22:36
    
@ZevChonoles I just deleted the question, did your comment somehow get re-directed here, or was that on purpose? –  Ethan Nov 2 '13 at 22:38
    
@Ethan: On purpose. –  Zev Chonoles Nov 2 '13 at 22:38
    
@ZevChonoles Quite off topic, but did you go to Chicago as an undergraduate? If so, I would greatly appreciate it if you could give me some advice on answering collegeadmissions.uchicago.edu/apply/essays. –  Ethan Nov 2 '13 at 22:44

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