Talk:Euler's continued fraction formula

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Field:  Analysis

Starting a new page

Hi, all!

Just a quick note to outline my intentions for this new article. I intend to present the formula the way Euler did (but in English, not Latin :), then state it in a more modern formulation, then indicate how it fits in with the articles convergence problem and fundamental inequalities, and cap it off with a few examples involving familiar power series (logarithm, exponential, maybe an arctan formula). DavidCBryant 23:19, 10 January 2007 (UTC)

Connection between complex logarithms and Pi

The continued fraction derived from the Mercator series for logz provides a cute continued fraction expansion for π when z = i. The algebra is pretty simple, but since it involves a value on the boundary of the circle of convergence for the series, I really should check around for some stuff (like Abel's test for convergence on the boundary) and reference that before sticking the example into this article. DavidCBryant 20:20, 12 January 2007 (UTC)

Irrationality of e

David, first of all I think it's a very nice page that you've made! Many of us either never learn about continued fractions (what with all those series and infinite products), or at any rate are not all that familiar with it, beyond perhaps periodicity of continued fractions for quadratic irrationalities, bad approximability of ${\displaystyle \phi ,}$ etc. I was actually looking for Euler's proof for irrationality of e: apparently, his was based on continued fractions, the now standard infinite series proof with factorials is due to Fourier is from a later period, 1815 or so. I can pretty much guess what it (Euler's proof) is from the continued fraction expansion of ${\displaystyle e^{z}}$ on the page, but if you have a copy of Introductio, as it appears, do you think you could check for it and include it (the proof)? It would seem like a nice application of the expansions. I suspect Lambert's proof of irrationality of ${\displaystyle \pi }$ was also based on the expansion that you gave. Arcfrk 06:12, 13 March 2007 (UTC)

Hi, Arcfrk. Thanks for the kind words. Sorry to be so slow to respond.
I don't have a copy of Euler's book. Perhaps I should have written the footnote differently – what I have is a book (Wall) that quotes a little bit of Euler's stuff.
I did do a bit of poking around, and I think Euler's proof that e is irrational was based on the regular continued fraction expansion for e, which is
${\displaystyle e=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{\ddots }}}}}}}}}}}}}$
or, in a more compact notation, e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...]. According to Szüsz and Rockitt, Euler put this expansion of e in his Introductio, but it's not clear that he (Euler) could actually prove it. Szüsz and Rockitt do give a proof based on the continued fraction expansion
${\displaystyle {\frac {e^{2/m}+1}{e^{2/m}-1}}=m+{\cfrac {1}{3m+{\cfrac {1}{5m+{\cfrac {1}{7m+{\cfrac {1}{\ddots }}}}}}}}.}$
They attribute this proof to Oskar Perron. When m = 2 a correspondence can be drawn between the convergents of this fraction and the 2nd, 5th, 8th, 11th, etc convergents of the regular continued fraction for e … it's kind of a pretty demonstration, so when I have the time I'll try to attach it to the article about e somehow. DavidCBryant 18:36, 4 April 2007 (UTC)

Error in "Notes"

The 4th note:

^ a b This series converges for |z| = 1, except when z = ±1, by Abel's test (applied to the series for log(1 − z))

${\displaystyle \log {\frac {1+w}{1-w}}={\cfrac {2w}{1-{\cfrac {w^{2}}{3-{\cfrac {4w^{2}}{5-{\cfrac {9w^{2}}{7-{\cfrac {16w^{2}}{\ddots }}}}}}}}}}\,}$