Fix a>0 and let x1>√(a). For n ≥1, define xn+1=(1)/(2)(xn+(a)/(xn)) Show that (xn) converges and

Step 1: **Show that the sequence u005C((x_n)u005C) is bounded below by u005C(u005Csqrt{a}u005C)**.u000Au000ASince u005C(x_1 u003E u005Cs

Fix a>0 and let x1>√(a). For n ≥1, define...

Fix $a>0$ and let $x_1>\sqrt{a}$. For $n \geq 1$, define
$$
x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right)
$$

Show that $\left(x_n\right)$ converges and that $\lim _{n \rightarrow \infty} x_n=\sqrt{a}$.

Key Concepts

Fixed Point Iteration

This concept involves finding a point x such that f(x) = x by repeatedly applying a function. In the context of iterative methods, the sequence generated by x??? = f(x?) is analyzed to see if it converges to a fixed point that satisfies the equivalent equation of the original problem.

Monotonicity and Boundedness

These are key properties in the study of sequences in real analysis. If a sequence is monotonic (either non-increasing or non-decreasing) and bounded, the Monotone Convergence Theorem guarantees that the sequence converges. Analyzing these properties is essential when proving the convergence of recursively defined sequences.

Newton’s Method

Newton’s method is an iterative technique for finding successively better approximations to the roots of a real-valued function. The recurrence relation in the question is a specific case of Newton’s method applied to the function whose zero corresponds to the desired square root, illustrating how iterative processes can efficiently converge to an accurate solution.

Fixed Point Analysis

Fixed point analysis is used to determine the limiting behavior of an iterative process by finding a value L that satisfies L = f(L). By setting the limit equal to the result of the iteration function, one can solve for the value, thereby confirming that it indeed satisfies the original problem’s equation.

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