Monotone sequence

**Connected** · April 3rd 2011, 04:28 PM

Given $y_n=\dfrac12(b+y_n^2)$ with $y_0=0.$

Show that $y_n$ is monotone.

I'm getting trouble to do that. It's a part of another problem which says to prove that $y_n$ converges, so in order to do that, I already proved that $y_n$ is bounded below.

**DrSteve** · April 3rd 2011, 07:01 PM

Do you mean $y_{n+1}=\dfrac12(b+y_n^2)$ ?

**Connected** · April 4th 2011, 06:18 AM

No, it's as is written.

Do you think there's a typo?

**DrSteve** · April 4th 2011, 06:47 AM

There must be. Otherwise the sequence is constant for n>0.

**Hartlw** · April 4th 2011, 07:03 AM

A constant sequence is monotonic. And Converges.

**DrSteve** · April 4th 2011, 09:47 AM

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.

**Hartlw** · April 4th 2011, 12:14 PM

yn is either a real or complex constant depending on b. in either case its constant, monotonic (absoluteley if b complex), and convergent.

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