4) Suppose x0=23,y0=3, xn=xn−1+yn−12xn−1yn−1 and yn=xnyn−1 for all n∈N. Prove that (a) xn↓x and yn↑y as n
Posted: Thu Jul 14, 2022 4:04 pm
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4) Suppose x0=23,y0=3, xn=xn−1+yn−12xn−1yn−1 and yn=xnyn−1 for all n∈N. Prove that (a) xn↓x and yn↑y as n→∞ for some x,y∈R; (b) x=y and 3.14155<x<3.14161. (x is actually π ) Note: If (xn) is monotonically decreasing (resp. montonically increasing) and converges to x then we write xn↓x (resp. xn↑x ). Thus, proving this requires the use of the Monotone Convergence Theorem.