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 an example of divergence?
editThe 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 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)
- Perhaps it's worth nothing that following that convention, we also neglect many "interesting" examples. Say . Can converge to 1, and still result in the product converging to 0? If not, then what is the highest number can approach? Forgive me if these are well known and simple problems to solve, it's new territory for me.
- Why should it be deemed that non-trivially converging-to-0 infinite products are necessarily uninteresting? 2601:647:C900:B6C0:99ED:F4A6:FA90:CDFA (talk) 05:52, 16 December 2022 (UTC)
How does this fit in?
editIf F(x) can be represented as a product:
then define G(x)=1-F(x) and using the series expansion for the natural logarithm:
so that and
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)
- Ok, probably after almost 6 years this comment might have been forgotten, but let me explain why I don't think is a good idea to include such observation in the article. What you say is a trivial statement: if is a fixed number, then
- Since you are expanding , you are assuming that for every , and then you are just applying the above formula point-wise. Note that this is quite different from Weierstrass' formula (if you wish, the problem with your expression is that you write in terms of , so that is not very useful, just like is not very useful to say that -for example- ). Lucha (talk) 16:10, 28 February 2017 (UTC)