Question

find lim x_n (n--> inf) if it exists. here x_1=3 and x_(n+1)= 1/ (4-x_n)

Answer 1

unfortunately im not able to come up with a concrete proof, but the sequence converges to 0. The idea is that if x_1 = 3, then:\[x_n\ge x_{n+1}\] for all natural numbers n. So you have a monotonic decreasing sequence. Also, it should be apparent that:\[x_n> 0\] for all n. So its a decreasing sequence that is bounded from below. So it converges to 0.