# Calculation of Limit of a Repeating Continued Fraction

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:

$$a'=\cfrac1{1+\cfrac1{5+a'}}$$

(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)

\begin{align}&a_0<\cfrac{-5+\sqrt{45}}2\\&a_{n+1}=\cfrac1{1+\cfrac1{5+a_n}}<\cfrac1{1+\cfrac1{5+\cfrac{-5+\sqrt{45}}2}}=\cfrac{-5+\sqrt{45}}2\end{align}

Induction on 2)

\begin{align}&a_1>a_0\\&a_{n+1}=\cfrac1{1+\cfrac1{5+a_n}}>\cfrac1{1+\cfrac1{5+a_{n-1}}}=a_n\end{align}

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$./indefCycle 1 5 -5 0 form 1 5 -5 1 0 0 1 To Return 1 0 0 1 0 form 1 5 -5 delta -1 ambiguous 1 form -5 5 1 delta 5 ambiguous 2 form 1 5 -5 form 1 x^2 + 5 x y -5 y^2 minimum was 1rep x = 1 y = 0 disc 45 dSqrt 6 M_Ratio 36 Automorph, written on right of Gram matrix: -1 -5 -1 -6 =========================================  • This looks very much like some vague code with many ambiguous steps. Care to explain? – Simply Beautiful Art May 29 '17 at 11:06 • @Simply I gave a fairly good tutorial in mathoverflow.net/questions/22811/… The word "ambiguous" means that, in the given coefficients$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 – 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. • Proving the sequence is monotone requires more than just$a_n<\frac{-5+\sqrt{45}}2\$. See my last hint for more details. – Simply Beautiful Art May 28 '17 at 21:35