Suppose we have the infinite product $2(1/2)(2^4)(1/2^8)(2^{16})\dots$ I have a hunch that the infinite product is $0$ despite partial product being strictly positive. Am I correct? If so, then how?
Thank you ahead of time!
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The part of the sentence "despite partial product being strictly positive" indicates that you are a tad surprised that the infinite product could be equal to zero? That's not so strange, consider $(1/2)(1/2)(1/2)(1/2).....$. All terms are strictly positive but the product has a limit zero. In your example, as indicated by others, the answer turns out to be a non zero value – imranfat Sep 20 '16 at 14:07
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Taking $\log_2$ of the sequence, we get that this limit is just $2$ to the power of
$$1-1+4-8+16$$
which does not converge in the normal sense, so the original product doesn't converge.
Carl Schildkraut
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2The even partial sums go to $-\infty$ and the odd partial sums go to $+\infty$. So it doesn't converge. For the product itself: the even partial products go to $0$ and the odd partial products go to $+\infty$. – GEdgar Sep 20 '16 at 14:03
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Note that we have: $2 \cdot \frac 12 \cdot 2^4 \cdot \frac{1}{2^8} \cdot... = 2 \cdot 2^{-1} \cdot {2^4} \cdot 2^{-8} \cdot ... = 2^{1-1+4-8+16-...}$
As the sum of the exponents doesn't converge we have that the product doesn't converge too.
Stefan4024
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For a limit of an infinite product to exist, you need the terms in the product to converge to the multiplicative identity (like with sums, to the additive identity). In your product they do not.
operatorerror
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1The terms can also converge to $0$. They don't, of course, but if they did it woul also be OK. – 5xum Sep 20 '16 at 14:03
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@5xum I don't see how? What happens if you take the log of the partial products? – operatorerror Sep 20 '16 at 14:04
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1We sometimes say that an infinite product "diverges to zero". So that could be the content of the question. In this case it does not diverge to zero. – GEdgar Sep 20 '16 at 14:05
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I see. I have seen the formulation that a product with terms of the form $1+a_n$ converges iff. the sum $\sum a_n$ converges which it would not if $a_n\rightarrow -1$ – operatorerror Sep 20 '16 at 14:06
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1The idea of saying that an infinite product diverges to zero is a consequence of the relationship with sums and the unfortunate convention of using the word "diverge" for limits approaching $-\infty$. – Sep 20 '16 at 14:39
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@qbert That formulation is true if the sequence $a_n$ is not arbitrarily close to $0$. Obviosly, the infinite product $0\cdot 0\cdot 0\cdots$ converges, and in fact the product converges if only one of the terms is $0$ (since, then, partial products are all equal to $0$). – 5xum Sep 21 '16 at 09:18