What Are the Limits of {xn} and {yn} in the Given Sequence?

[huey910]

Given x0 and y0 such that x0 > y0 > 0. Define, for n = 0,1,2,,
xn+1 =xn +yn , yn+1 = 2xnyn .Find the limits of {xn} and {yn}.

why is the answer lim{xn} = lim{yn} = sqrt(x0y0)?

 

[mathman]

The problem as stated is wrong (e.g. x0 = 2, y0 =1. they just keep getting bigger).

 

[huey910]

Oh! Sorry, I transcribed the question erroneously. It should be:

Given x0 and y0 such that x0 > y0 > 0. Define, for n = 0,1,2,,
xn+1 =(xn +yn)/2 , yn+1 = 2xnyn/(xn + yn) .Find the limits of {xn} and {yn}.

why is the answer lim{xn} = lim{yn} = sqrt(x0y0)?

 

[disregardthat]

1) Show that x_n is decreasing, and y_n is increasing.
2) Show that x_n has a lower bound, and y_n has an upper bound.

Hint: Show that x_n < y_n by induction, and then that x_n+1 < x_n, and y_n+1 > y_n.

Do you know why they must converge in this case? It is a well known fact.

3) Find the limits by using the equations.

 

[huey910]

It is a well-known fact? Please expand on it and enlighten me! Thank you for the hints though!