Consider The Iteration. Show That X_n Decreases Monotonically To √2. Is The Rate Linear Superlinear Or Quad?

1. It is not true that x_n decreases monotonically to √2. It depends on x_0 and x_1.
A. if x_1 = x_0 = 0, then x_2 is undefined
B. if x_1 = 1, x_0 = 0, then x_2 = 2, which is an increase
2. If both x_n and x_(n-1) are close to but greater than √2
then x_(n+1) = x_n – dx, where dx is positive, so x_(n+1) < x_n
Furthermore, if x_n = √2 + e, then dx is approximately:
2e / (2 √2) or e / √2
Because √2 > 1, e / √2 = dx is less than e, so x_(n+1) > √2
As for convergence, if the denominator were 2 x_n, this would just be Newton’s method, which converges quadratically, but since the denominator is x_n + x_(n-1) it does not converge quadratically.