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The given sequence comes from the recursion formula of Newton method

$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$ I am given $x_0=1, x_{n+1}=x_n-\frac{x^2_{n}-2}{2x_n}$

I need to show the value to which it converges . I have no idea how to proceed . Please provide some hints.

I only can understand that the function generating this sequence is $x^2-2$. I don't know how to proceed after that.

The Learner
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    If it converges to something, say $a$, then $a=a-\frac{a^2-2}{2a}$ by continuity. This is how to find the value. Of course you also need to show that it converges to something. – GEdgar Nov 23 '19 at 18:36
  • You need to get to the root of the issue. – copper.hat Nov 23 '19 at 18:52
  • Got the answer. Thanks mates – The Learner Nov 23 '19 at 19:26
  • @copper.hat can you please suggest me 1-2 books so that I can answer topics related to sequences and series AND matrices and determinants. I am highly interested in these two topics sir. I want to answer the questions posted on this site related to this topic . – The Learner Nov 23 '19 at 19:30
  • @TheLearner: I will give you the names of two books I like, but you should keep in mind that book choice & how one reacts to them is very personal and it has been many decades since I first dealt with such topics so my recollection of what was most useful back then has dimmed quite a bit. – copper.hat Nov 23 '19 at 19:35
  • @TheLearner: For matrices, etc, I like "Matrix Analysis" by Golub & Van Loan, and for sequences (in particular with regard to Newton's method), I like Kantorovich & Akilov's "Functional analysis". Unfortunately they may be hard or expensive to get now. – copper.hat Nov 23 '19 at 19:38
  • Thanks a lot for the suggestions sir. – The Learner Nov 23 '19 at 20:04
  • Sir the name of the book by Golub amd van Loan is Matrix computations – The Learner Nov 23 '19 at 20:09

2 Answers2

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Write $$x_{n+1}=\frac{1}{2}\left(x_n+\frac{2}{x_n}\right)\geq \sqrt{2}$$ by AM-GM.

Dr. Sonnhard Graubner
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Suppose $f$ is continuously differentiable.

If $x_n$ is generated by Newton's method and $x_n \to x$ and $f'(x) \neq 0$, then by continuity we must have $x = x-{f(x) \over f'(x)}$ from which it follows that $f(x)=0$.

In the above case, $f(x) = x^2-2$ and so, assuming it converges, you must have $x = \pm \sqrt{2}$.

copper.hat
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