# Prove $(x_n)$ defined by $x_n= \frac{x_{n-1}}{2} + \frac{1}{x_{n-1}}$ converges when $x_0>1$

$x_n= \dfrac{x_{n-1}}{2} + \dfrac{1}{x_{n-1}}$

I know it converges to $\sqrt2$ and I do not want the answer. I just want a prod in the right direction.

I have tried the following and none have worked: $x_n-x_{n-1}$ and this got me nowhere.

I have tried $\dfrac{x_{n+1}}{x_{n}}$ with no luck, and I was expecting this to help,

• Prod: Plot the graphs of the function $x\mapsto\frac{x}2+\frac1{x}$ and $x\mapsto x$ on the same diagram. – Did May 7 '16 at 5:37
• So I have done this and found they intersect at $\sqrt2$ in the 1st quadrant. – stackdsewew May 7 '16 at 5:44
• Right, and now did you visualize a sequence $(x_n)$ on the graph thanks to the usual ladder-like construction? What does this suggest? – Did May 7 '16 at 5:46
• I fail to see any ladder-like construction... Anyway, it should be clear that after at most one iteration, $x_n\geqslant\sqrt2$ and that, after that, $(x_n)$ is nonincreasing. Can you show these? Once this is done, a natural idea is to compute $x_{n+1}-\sqrt2$ in terms of $x_n-\sqrt2$ and possibly of other quantities and to see what happens. Did you do that? Another natural idea is to check that $(x_n)$ is decreasing and to conclude directly from this as the answerer below did. Actually, the whole point of the graph is to suggest tons of properties of $(x_n)$ to prove to get the conclusion. – Did May 7 '16 at 5:53
• Note that this is the Newton iteration for the square root $x_n=\frac12(x_{n-1}+a/x_{n-1})$, with $a=2$. – Yves Daoust May 7 '16 at 13:21

## 3 Answers

We have: By AM-GM inequality: $x_n > 2\sqrt{\dfrac{x_{n-1}}{2}\cdot \dfrac{1}{x_{n-1}}}=\sqrt{2}, \forall n \geq 1$. Thus: $x_n-x_{n-1} = \dfrac{1}{x_{n-1}} - \dfrac{x_{n-1}}{2}= \dfrac{2-x_{n-1}^2}{2x_{n-1}} < 0$. Hence $x_n$ is a decreasing sequence,and is bounded below by $\sqrt{2}$. So it converges, and you showed the limit is $\sqrt{2}$.

Observe that

$$\frac{x_n-\sqrt2}{x_n+\sqrt2}=\frac{\dfrac{x_{n-1}}{2} + \dfrac{1}{x_{n-1}}-\sqrt2}{\dfrac{x_{n-1}}{2} + \dfrac{1}{x_{n-1}}+\sqrt2}=\frac{x_{n-1}^2-2\sqrt2x_{n-1}+2}{x_{n-1}^2+2\sqrt2x_{n-1}+2}=\left(\frac{x_{n-1}-\sqrt2}{x_{n-1}+\sqrt2}\right)^2.$$

Then by induction,

$$\frac{x_n-\sqrt2}{x_n+\sqrt2}=\left(\frac{x_0-\sqrt2}{x_0+\sqrt2}\right)^{2^n}.$$

Then whenever

$$\left|\frac{x_0-\sqrt2}{x_0+\sqrt2}\right|<1$$ (in particular $x_0>0$) the series converges to $\sqrt2$.

Let $f(x)=x^2-2$, note that $f$ is convex and hence it is straightforward to show that if $x_n$ is the Newton iteration for the equation $f(x) = 0$, and $x_0>0$, then $x_n \ge \sqrt{2}$ (equivalently, $f(x_n) \ge 0$) if $n \ge 1$, and $\sqrt{2} \le x_{n+1} \le x_n$ for $n \ge 1$.

• Sorry if it is a silly question but how does your last inequality prove convergence Is it similar to a squeeze theorem like principle? – stackdsewew May 7 '16 at 13:36
• @anna_xox: The same was as in the accepted answer, for $n \ge 1$, the sequence $x_n$ is non increasing and bounded below by $\sqrt{2}$, hence it converges to some $x$. Since $f(x) = 0$, we see that $x= \sqrt{2}$. – copper.hat May 7 '16 at 22:27