如何证明a1=1,an+1=3-1/an这个数列收敛并求极限

Question

如何证明a1=1,an+1=3-1/an这个数列收敛并求极限

Answer 1

consider
f(x) = 3- 1/x
f'(x) = 1/x^2 >0
f(x) is increasing
a2 = 3-1/a1= 2 >a1
=> {an} is increasing
|an| lim(n->∞)an exists
let
L = lim(n->∞)an
a(n+1)=3- 1/an
L = 3- 1/L
L^2- 3L +1=0
L = (3+√5)/2
lim(n->∞)an = L = (3+√5)/2