+0  
 
0
283
2
avatar+95 

Find the value of

\(\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{2+...}}}}\)

benjamingu22  Jun 5, 2017
 #1
avatar+87301 
+2

Let  x  =                  1        

                          1 +    1    

                                  2 + ....

 

So  we have that 

 

x   =     1 / [ 1 +  1/ ( 2 + x) ]

 

x  =  1(2 + x) / (2 + x + 1)

 

x =  (x + 2) / (x + 3)

 

x (x + 3)  =  x + 2

 

x^2 + 3x = x + 2

 

x^2 + 2x - 2  = 0

 

Taking the positive solution, x = √3  -  1

 

 

 

cool cool cool   

CPhill  Jun 5, 2017
 #2
avatar+95 
+1

Thank you, CPhill.

benjamingu22  Jun 8, 2017

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