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?


Let $a_0=0$ and $a_{n+1}=\cfrac1{1+\cfrac1{5+a_n}}$.

Prove by induction:

1) $a_n<\dfrac{-5+\sqrt{45}}2$

2) $a_{n+1}>a_n$ i.e. monotone

So that it converges and that it must converge to some $a'$ such that:


(feel free to ask if you need more tips on any steps below, hover on the below tips to see major steps)

Induction step on proving 1)


Induction on 2)



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 $t^2 + 5t-5.$

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
  • $\begingroup$ This looks very much like some vague code with many ambiguous steps. Care to explain? $\endgroup$ – Simply Beautiful Art May 29 '17 at 11:06
  • 1
    $\begingroup$ @Simply I gave a fairly good tutorial in mathoverflow.net/questions/22811/… The word "ambiguous" means that, in the given coefficients $a,b,c$ of the implied form $a x^2 + b xy + c y^2,$ we have $a | b.$ Details in that post, also Buell's book, also L. E. Dickson Introduction to the Theory of Numbers $\endgroup$ – Will Jagy May 29 '17 at 16:14

Hint : Simply consider the difference between two successive recurring terms and then it is easy to show that the sequence is monotonically increasing.
Now, $$\frac{1}{5} > 0\\\implies1+ \frac{1}{5} > 1\\\frac{1}{1 +\frac{1}{5}} < 1$$.

Use this argument several times and try to deduce that the given sequence is bounded above by 1 and then by monotone-bounded theorem, you know that there exists a limit of the sequence. Let it be l. Now a routine computation yields your answer.

N.B.: in order to show that the sequence is monotone , I guess you will need to show by induction that $$a_n < \frac{-5 + \sqrt{45}}{2}$$. And this can also come handy to show that the sequence is bounded above.

  • $\begingroup$ Proving the sequence is monotone requires more than just $a_n<\frac{-5+\sqrt{45}}2$. See my last hint for more details. $\endgroup$ – Simply Beautiful Art May 28 '17 at 21:35

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