Talk:Convergent (continued fraction)

Text and/or other creative content from Convergent (continued fraction) was copied or moved into Generalized continued fraction with this edit. The former page's history now serves to provide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists.

Partial numerators and denominators

Hi, Dick! Thank you for creating this stub.

I'm going to add just a little bit to this article right now, about the partial numerator and the partial denominator. I'm also working on the quadratic irrational article, and will probably get around to some other topics in the theory of continued fractions fairly soon. DavidCBryant 19:32, 29 November 2006 (UTC)[reply]

You're welcome. I just removed the partial numerator and partial denominator stuff, since they remained undefined. I didn't see your note here until after I did that. If you add articles on them, or even better, local definitions, put them back. Dicklyon 20:04, 29 November 2006 (UTC)[reply]

Yeah, well, after I wrote that in I realized (10 minutes later) that I had used the terms incorrectly. So you did me a favor by taking that out. DavidCBryant 20:32, 29 November 2006 (UTC)[reply]

Convergents and convergence

I just added a couple of headings to this stub article, plus a little about the concept of a convergent continued fraction, as that term is used in analysis. I noticed that there's a disambiguation page for divergent, which also needs to be fixed up. Since the article generalized continued fraction is sort of a mess and there's nothing convenient to hang my hat on there, I probably ought to put the fundamental recurrence formulas in their own article, too. I've also got a couple of convergence examples that I'll add to this article soon. DavidCBryant 17:50, 1 December 2006 (UTC)[reply]

Isn't convergent a noun in this context. I believe the finite fractions [a0], [a0;a1], [a0;a1;a2], etc. are known as convergents of the partial fraction. Note, I haven't looked this up. I'm going by memory here. Greg Woodhouse 22:35, 20 March 2007 (UTC)[reply]

Well, the title of the article refers to a convergent, so that's a noun. And that's what the article is about – the continued fraction (with complex partial numerators and denominators, as in analysis) converges if the sequence of convergents approaches a limit (which may be the point at infinity, in the most general discussions, where values are taken from the extended complex plane).

From the notation you've used – [a₀; a₁, a₂, …] – it appears that you're thinking of regular continued fractions (where all the partial numerators are unity, and the partial denominators are positive integers). Those are interesting, but they're not nearly as much fun as the ones with complex elements. DavidCBryant 23:42, 20 March 2007 (UTC)[reply]