Infinite Continued Fraction

Date: 05/15/2002 at 01:26:16
From: Jaimie
Subject: infinite continued fraction

I need your help to figure out this question from my school. I spent 
hours and hours, but I remain clueless. Here is the question:

What can you determine about the value of the infinite continued 
fraction [1;1,2,3,1,2,3,1,2,3....]?

This continued fraction may also be written;

     1
1+_________
       1
  2+_________
         1
    3+_________
           1
      1+________

        1+.......

How do you do this problem?

Date: 05/15/2002 at 09:11:16
From: Doctor Paul
Subject: Re: infinite continued fraction

Hi,

Let x = [1, 2, 3, 1, 2, 3, ...]

Then 

         1
 x = -----------
         x1

where x1 = [2, 3, 1, 2, 3, ...].  (That is, x1 is x with the first 
element removed.) 

Continuing, 

          1
  =  -----------
          1
     2 + -----
          x2

where x2 = [3, 1, 2, 3, 1, 2, 3, ...], and 


          1
  =  ---------------- 
              1
     2 + ------------
                 1
           3 + ------
                 x3

where x3 = [1, 2, 3, 1, 2, 3, ...] = x.  So we can write

          1
 x = ---------------- 
              1
     2 + ------------
                 1
           3 + ------
                 x

This can be solved for x.  You fill in the details.  You'll get a 
quadratic equation in x.  The answer cannot be negative (since all 
terms in the continued fraction are positive) so you know to choose 
the positive root.  You'll end up with:

       4 + sqrt(37)
  x = -------------
           7

I hope this helps.  Please write back if you'd like to talk about 
this some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/