# Monotone sequence

• Apr 3rd 2011, 04:28 PM
Connected
Monotone sequence
Given \$\displaystyle y_n=\dfrac12(b+y_n^2)\$ with \$\displaystyle y_0=0.\$

Show that \$\displaystyle y_n\$ is monotone.

I'm getting trouble to do that. It's a part of another problem which says to prove that \$\displaystyle y_n\$ converges, so in order to do that, I already proved that \$\displaystyle y_n\$ is bounded below.
• Apr 3rd 2011, 07:01 PM
DrSteve
Do you mean \$\displaystyle y_{n+1}=\dfrac12(b+y_n^2)\$?
• Apr 4th 2011, 06:18 AM
Connected
No, it's as is written.

Do you think there's a typo?
• Apr 4th 2011, 06:47 AM
DrSteve
There must be. Otherwise the sequence is constant for n>0.
• Apr 4th 2011, 07:03 AM
Hartlw
A constant sequence is monotonic. And Converges.
• Apr 4th 2011, 09:47 AM
DrSteve
This particular sequence may or may not be monotonic. The given equation may have 2 solutions (depending on b), and therefore the given sequence isn't well defined - it could take on one of these values, or bounce between the 2. In any case, it is more likely that the problem has a typo.
• Apr 4th 2011, 12:14 PM
Hartlw
yn is either a real or complex constant depending on b. in either case its constant, monotonic (absoluteley if b complex), and convergent.