Answer to Question #280116 in Real Analysis for abc

Question

Check the convergence of the sequence defined by 𝑢𝑛+1 = (1 + 1/ 𝑢𝑛 ) , 𝑢1 > 0. Note that this is the sequence associated with the continued fraction expansion of the Golden ratio. 

Answer 1

Solution:

Given sequence, $u_{n+1}=1+\dfrac1{u_n},u_1>0$

Let the sequence is convergent to $l$.

$\therefore l=1+\dfrac 1l \\ \Rightarrow l^2=l+1 \\ \Rightarrow l^2-l-1=0 \\ \Rightarrow l=\dfrac{1\pm\sqrt5}{2} \\ \because u_1>0 \\\therefore l=\dfrac{1+\sqrt5}{2}$

Hence, the given sequence is convergent to $\dfrac{1+\sqrt5}{2}$ .