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Find the value of\cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \dotsb}}}}

difficulty advanced
Guest Jan 31, 2015

Best Answer 

 #6
avatar+889 
+5

Successive iterates are 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/56, 112/153, 153/209, 418/571, 571/780 ...

They are converging to sqrt(3) - 1 = 0.7320508... .

The last one that I've listed, 571/780 = 0.732051 6dp.

Bertie  Feb 1, 2015
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8+0 Answers

 #1
avatar+889 
+5

Put the continued fraction equal to x, and make the substitution further into the fraction to get

$$x=\frac{1}{1+\frac{1}{2+x}}}$$ 

Tidying that up gets you quadratic $$x^{2}+2x-2=0,$$ from which $$x=\sqrt{3}-1.$$

Bertie  Feb 1, 2015
 #2
avatar+85753 
0

Very nice, Bertie....!!!

 

CPhill  Feb 1, 2015
 #3
avatar+92217 
+5

thanks Bertie,

I did not have a clue what to make of this question.    

 

you got a mention in my NewYear 2015 post  

http://web2.0calc.com/questions/happy-new-year-everyone_1

Melody  Feb 1, 2015
 #4
avatar+92217 
0

I do not understand this at all.   

Melody  Feb 1, 2015
 #5
avatar+85753 
+5

Bertie....let me ask you a question about this one.....when I carry it out for a few iterations, I get that the results seem to "oscillate' between both roots of that quadratic, i.e., sqrt(3)±1

Did I do something wrong???

CPhill  Feb 1, 2015
 #6
avatar+889 
+5
Best Answer

Successive iterates are 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/56, 112/153, 153/209, 418/571, 571/780 ...

They are converging to sqrt(3) - 1 = 0.7320508... .

The last one that I've listed, 571/780 = 0.732051 6dp.

Bertie  Feb 1, 2015
 #7
avatar+85753 
0

OK, thanks....

 

CPhill  Feb 1, 2015
 #8
avatar+92217 
0

CPhill just sent me this on continued fractions for anyone who is interested :)

http://webserv.jcu.edu/math//vignettes/continued.htm

 

Thanks CPhill  

Melody  Feb 1, 2015

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