Talk:Infinite product

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This article is probably very non-ideal, but when I needed this material a while ago it was hard to find, and so I figured wikipedia would be a good place to hold it. Please feel free to change it around and make it more encyclopedia-like, even more so than usual. :)


We probably need a little caveat about negative numbers: the log formula won't work without problems in that case. Maybe use a different log which is defined on the negative reals? --AxelBoldt

A product of negative coefficients does not converge; it can at most flip. --Yecril (talk) 18:00, 20 December 2008 (UTC)

Inaccurate Statement[edit]

"Therefore, the logarithm log an will be defined for all but a finite number of n" neither makes sense nor true as I understand it. One can easily imagine a sequence which converges to 1 that is not finite. one simple example is a(n)= 1+(1/n) where n is an integer. This of course is not limited for a being real, for example think of any oscillating function which dampens to 1. this might have been a simple typo the author made, but i don't understand his/her intention, so i am timid to edit this —Preceding unsigned comment added by 129.49.61.161 (talk) 17:06, 17 August 2010 (UTC)

Maybe it is a left-over from careless editing. Some sources allow a finite number of 0 factors and still call the product convergent if the product of the non-zero factors is non-zero. Probably that is the meaning. I'll try to fix it. McKay (talk) 06:09, 31 March 2011 (UTC)

Why is \prod_{k=1}^{\infty} a_k = \lim_{n\rightarrow\infty} \prod_{k=1}^{n} a_k = 0 an example of divergence?[edit]

The article currently indicates that if the infinite product is zero then that is an example of divergence. Why? I would think that that is an example of convergence, despite that a corresponding sum \sum_{k=1}^{\infty} \log(a_k) is not finite. —Quantling (talk | contribs) 18:35, 30 March 2011 (UTC)

It is just a convention. The reason for the convention is that allowing convergence to 0 would admit too many uninteresting series (for example, any sequence {an} with | an| < 1/2), and some desirable properties would disappear. McKay (talk) 06:15, 31 March 2011 (UTC)

How does this fit in?[edit]

If F(x) can be represented as a product:

F(x)=\Pi_{n=1}^\infty f_n(x)

then define G(x)=1-F(x) and using the series expansion for the natural logarithm: \ln(1-G(x))= -\sum_{n=1}^\infty G(x)^n/n

\ln(F(x))=-\sum_{n=1}^\infty G(x)^n/n = \sum_{n=1}^\infty \ln(f_n(x))

so that f_n(x)=e^{-G(x)^n/n} and

F(x)=\Pi_{n=1}^\infty\, e^{-\frac{1}{n}(1-F(x))^n}

I have checked this on Mathematica for a number of functions, and Mathematica agrees, but I have no idea what the conditions are for this formula to work. Can anyone find a reference for this and put a section into the article? Its a very useful and general expression, and much simpler than the more general Weierstrass development. PAR (talk) 17:45, 13 June 2011 (UTC)