Talk:Fundamental recurrence formulas

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Working backwards[edit]

If we arbitrarily set a₀=1, then we can extend the recurrence relations backwards as is done for regular continued fractions:

{\begin{aligned}h_{-2}&=0&k_{-2}&=1\\h_{-1}&=1&k_{-1}&=0\\h_{0}&=b_{0}&k_{0}&=1\\h_{1}&=b_{1}b_{0}+a_{1}&k_{1}&=b_{1}\\h_{n+1}&=b_{n+1}h_{n}+a_{n+1}h_{n-1}&k_{n+1}&=b_{n+1}k_{n}+a_{n+1}k_{n-1}\,\end{aligned}}

What do you think? JRSpriggs 09:21, 16 December 2006 (UTC)[reply]

Looks good except that a₀ is not defined. That's why it's not mentioned in the recurrence formulas. Glenn L (talk) 22:53, 2 May 2009 (UTC)[reply]

Convergence Problem[edit]

In the section The determinant formula the account of "the most obvious direct attack on the convergence problem" is misleading. The terms of an infinite series are the differences between consecutive partial sums and for the series (or equivalently the sequence of partial sums) to converge it is not sufficient that the terms approach zero, e.g the harmonic series. Unless anyone objects or pre-empts me I propose to recast this paragraph. --Ian S (talk) 16:24, 2 May 2009 (UTC)[reply]

golden ratio[edit]

Since all the partial numerators in this example are equal to one, the determinant formula assures us that the absolute value of the difference between successive convergents approaches zero quite rapidly.

The golden ratio has been called "the most irrational number" because its c.f. converges so slowly! The "assurance" means nothing without some knowledge of the behavior of $B_{n}B_{n-1}$ . —Tamfang (talk) 05:19, 27 October 2010 (UTC)[reply]

Looks like I confused numerators with denominators. —Tamfang (talk) 22:19, 20 June 2023 (UTC)[reply]

Suggest merger[edit]

I suggest merging this back into either Generalized continued fraction or Convergent (continued fraction). There seem to be several articles covering the material already, and this one is unsourced and with a debatable name. Deltahedron (talk) 09:48, 18 November 2012 (UTC)[reply]

Support I'd merge both this and the convergent article into Generalized continued fraction and replace them with redirects. They aren't separate topics and they don't have anything particular extra in them. Dmcq (talk) 17:04, 14 January 2013 (UTC)[reply]

I recommend discussing this at Talk:Generalized_continued_fraction#Merge_discussion, where there are likely to be more eyes. RockMagnetist (talk) 17:51, 14 January 2013 (UTC)[reply]