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### 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
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/
```
Associated Topics:
College Number Theory
High School Number Theory
High School Sequences, Series

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