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Suppose $\{ \alpha_n \}$ is a decreasing sequence of real numbers such that $0 < \alpha_n < 1$ and $\alpha_n$ goes to $0$ as $n$ goes to infinity.

I was wondering if there is a known condition for $\{ \alpha_n \}$ so that the product $\prod (1- \alpha_n)$ will not be $0$?

Thanks!

Dominic Michaelis
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Tom Mosher
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1 Answers1

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Everytime when the Product converges the limit won't be zero, because per definition a infinite product only converges when it limit is not zero. Using that

$$\log\Big( \prod_{i=1}^n a_i \Big) = \sum_{i=1}^n \log(a_i) $$ one just can use the well known results for series to test the convergence of products.

Dominic Michaelis
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  • Is that really the definition of a product converging? Isn't it more natural to define the product as converging if and only if the partial products converge? – Thomas Andrews Jun 03 '13 at 14:43
  • @ThomasAndrews In fact that seems more natural, but this gives a lot of problems, because of if one factor is zero the hole product will always be zero, and so the convergence will heavily depend on a single factor, which is a really ugly thing. So the Definition is that a product converges only if the limit of the partial products converge and is not zero. This leads to the possibility going back to sums for checking convergence – Dominic Michaelis Jun 03 '13 at 14:46
  • Obviously, if one factor is zero, that's true, but it can converge to zero even if all the factors are non-zero. – Thomas Andrews Jun 03 '13 at 14:50
  • @ThomasAndrews If we allow convergence to zero, products over unbounded sequences could converge, take for example $a_{2k}=\frac{1}{2k}$; $a_{2k+1}=k$ which gives you something like the conditional convergent for series – Dominic Michaelis Jun 03 '13 at 15:02
  • Yes, but the I'm not clear what the point of that comment is. The question above clearly doesn't allow for this, while it does allow for convergence to zero. – Thomas Andrews Jun 03 '13 at 15:11
  • @ThomasAndrews Sorry because I don't get your point too. Everytime when the product converges according to definition of wikipedia or conway the limit will be not zero and we can use our standard tests for sums. In the last comments I tried to explain why it makes sense not to allow $0$ as the limit of a product – Dominic Michaelis Jun 03 '13 at 15:17
  • Matter of taste, I guess, but when answering a question, "when will this infinite product not be zero," it struck me as more confusing than helpful to say, "when it converges, it is not zero, by definition." That didn't get to the heart of the question, but I guess if OP accepted your answer, he is okay with it. :) – Thomas Andrews Jun 03 '13 at 15:22
  • @ThomasAndrews The standard term for that behavior in analysis is actually divergence to 0, and the reason that that phrase is used is because of the analogy with infinite sums like the harmonic sum. – Steven Stadnicki Jun 03 '13 at 15:57