I need to show the following:

Starting with a point $S=(a,b)$ of the plane with $a>b>0$, we generate a sequence of points $(x_n,y_n)$ according to the rule: $$x_0=a, y_0=b, x_{n+1}=\frac{x_n+y_n}{2}, y_{n+1}=\frac{2x_ny_n}{x_n+y_n}.$$ Prove that $\lim x_n=\lim y_n=\sqrt{ab}$.

I can show that $x_ny_n=ab$ and $x_n>y_n$ for all $n$, but I am having trouble proving $\lim x_n=\lim y_n=\sqrt{ab}$. Suggestions?

  • $\begingroup$ First prove convergence. Then use the recurrence to find the limit. $\endgroup$ – J.R. Feb 16 '14 at 18:29

Let us check that $(x_n)_n$ is a convergent sequence. It is monotonously decreasing:


On the other hand the sequence is bounded from below: $x_{n}\ge 0$.

Therefore it converges to some limit $x=\lim_{n} x_n$ (monotone convergence theorem). Since $x_ny_n=ab$ for all $n$, also $(y_n)_n$ converges to some limit $y=\lim_n y_n$. Now the recurrence for $x_n$ implies


That is, $x=y$. Since $xy=ab$, we have $x=y=\sqrt{ab}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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