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Some help with a continued fraction?

  1. Mar 9, 2009 #1
    I wasn't exactly sure where to put this, so I decided to try this section. It may be more appropriate to put this in the Calculus / Analysis section... I don't know.

    Can anybody tell me whether or not it's possible to find the value of the continued fraction

    0 + 1 / (2 + 3 / (4 + 5 / (6 + 7 / (...))))

    I wrote a relatively simple program to calculate partial fractions... when I go up to the nth odd number, I get this sequence:

    1
    0.2
    0.428571
    0.372549
    0.38057
    0.379654
    0.379738
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732

    When I go up to the nth even integer, I get:

    0
    0.5
    0.363636
    0.381579
    0.379562
    0.379745
    0.379731
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732
    0.379732

    So it looks like it converges... to something.

    Does anybody know of a way to (a) determine whether it actually converges to anything and (b) find this value analytically, in closed form?

    Just a curiosity...

    P.S. I know that most continued fractions like this don't have a closed form, I just thought this one was of interest because it seems so simple... just the numbers, you know.

    Thanks!
     
  2. jcsd
  3. Mar 9, 2009 #2
    Last edited by a moderator: Apr 24, 2017
  4. Mar 9, 2009 #3
    Nice...
     
  5. Mar 10, 2009 #4

    CRGreathouse

    User Avatar
    Science Advisor
    Homework Helper

    There's a comment on Sloane's http://www.research.att.com/~njas/sequences/A113014 [Broken] that the constant is equal to

    sqrt(2e/pi)/erfi(1/sqrt(2)) - 1
     
    Last edited by a moderator: May 4, 2017
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