Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $x_1 > 0$ and

$ x_{n+1} = 1/2 (x_n + 2/x_n)= x^2n+2/2x_n, n\geq1.$

Does $\{x_{n+1}\}$ converge? If so, find its limit.

Hint: Prove first that if $a, b \geq 0$ ,then $ 2ab \leq a^2 + b^2. $

I don't see how the hint is suppose to help.

share|improve this question
Try formatting your expressions to avoid ambiguity, in particular your fractions. –  glebovg Nov 9 '12 at 4:33
Your equality does not make sense. Please edit. –  glebovg Nov 9 '12 at 4:37

1 Answer 1

up vote 0 down vote accepted

Suppose it converges, then both ${x_n}$ and ${x_{n+1}}$ approach the same limit, say $x$. This means (if I deciphered your expression correctly) $$x = \frac{1}{2}\left( {x + \frac{2}{x}} \right)$$ or $$2{x^2} = {x^2} + 2$$ so $$x = \pm \sqrt 2.$$ Since all terms are positive, the limit must be $\sqrt 2$.

share|improve this answer
We can prove that $x_2\geq x_3\geq\ldots\geq\sqrt{2}$, so the sequence converges. –  Siming Tu Nov 9 '12 at 4:51

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.