Answer to Question #257363 in Real Analysis for saduni

Question

Check the convergence of the sequence defined by 𝑢_𝑛+1 =1/2 (𝑢_𝑛 + 𝑎 /𝑢_𝑛 ) , 𝑎 > 0.Note that this is the sequence associated with finding the square root of a number 𝑎 > 0 by the Newton’s method.

Answer 1

let the sequence $u_n$ converges to $l$ .

Then,

$u_{n+1}=\frac{1}{2}(u_n+\frac{a}{u_n})\\ l=\frac{1}{2}(l+\frac{a}{l})\\ \text{Solving this equation,we get}\\ l=\frac{1}{2}(\frac{a+l^2}{l})\\ 2l^2=a+l^2\\ l^2=a\\ l=\pm\sqrt{a}\\ \text{Since, it is given a>0. Therefore, }\\l=\sqrt{a}.$