Infinite Continued FractionDate: 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/ |
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