[Problem] Calculating the limit of 2 sequences

[assaf127]

Hi

I have this exercise, and Ididn't manage to complete it:

There are 2 sequences, An and Bn, defined as:
http://img517.imageshack.us/i/yyyx.jpg/ (I also attached it to this thread)
And: a>b>0

I need to prove that the sequences converge, and to calculate thier limit.

Now, I proved that they converge:
Because of the inequality of means, a>=An>Bn>=b, so An and Bn are bounded between a and b.
An monotonically decreases and Bn monotonically increases, so they both converge.
An=(An+Bn)/2
So when I calculate the limit of that when n goes to infinity:
La=(La+Lb)/2
2La=La+Lb
La=Lb
And I get the same result for the definition of Bn, so I can't find a way to calculate this limit (as a function of a and b).

Any ideas?

 

[galactus]

Notice that the one involving a is the arithemtic mean and the one involving b is the geometric mean

 

[assaf127]

I can't see where it can further help me than what I already did.. I need to find some kind of an "average" that will be the limit of of these sequences.. I thought about making a sequence, say Cn, where for every n: Bn<Cn<An, and then use sandwitch, but I couldn't find how to..

 

[galactus]

Together, they form the arithmetic-geometric mean.

See here. It may be a nice reference:

http://escholarship.org/uc/item/5kx0j9nm