Question #280116

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. 


1
Expert's answer
2021-12-20T16:45:49-0500

Solution:

Given sequence, un+1=1+1un,u1>0u_{n+1}=1+\dfrac1{u_n},u_1>0

Let the sequence is convergent to ll.

l=1+1ll2=l+1l2l1=0l=1±52u1>0l=1+52\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 1+52\dfrac{1+\sqrt5}{2} .


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