TITLE: Calculation of Limit of a Repeating Continued Fraction
QUESTION [9 upvotes]: I'm sorry if this is a duplicate. I have no idea on what kind of "name" i should give to this, and therefore i have no idea on how to search on the internet for help on understanding it. If it happens that this is a duplicate, i would be grateful if you could link me to where there are any solutions for this.
I need to prove for an exercise on my analysis book that the following sequence
$$
{\cfrac{1}{1+\cfrac{1}{5}}},\quad
{\cfrac{1}{1+\cfrac{1}{5+\cfrac{1}{1+\cfrac{1}{5}}}}},\dotsc
$$
is monotone and converges to ${\frac{-5+\sqrt{45}}{2}}$
I imagine that once i get on how to determine it's limit, it will be easy to prove that it is in fact monotone. I have no idea on how to approach it though. Any tips?

REPLY [4 votes]: Here is the method of Lagrange and Gauss for this: the "reduced" quadratic form $x^2 + 5x - 5 y^2$ is in a very short cycle. Notice that the limit is the positive root of $t^2 + 5t-5.$
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 1 5 -5

  0  form              1           5          -5


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   1 5 -5   delta  -1     ambiguous  
1  form   -5 5 1   delta  5     ambiguous  
2  form   1 5 -5


  form   1 x^2  + 5 x y  -5 y^2 

minimum was   1rep   x = 1   y = 0 disc 45 dSqrt 6  M_Ratio  36
Automorph, written on right of Gram matrix:  
-1  -5
-1  -6
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