Limit of Sequences help

[jai6638]

Hey there..

I have to find the limit of the sequence:

Sqrt(2), Sqrt(2*Sqrt(2)), Sqrt(2*Sqrt(2*Sqrt(2))),....

For an expression, I came up with

Sqrt(2n) but that doesnt work unless, n=1, n=Sqrt(2), n=Sqrt(sqrt2))...

However, not sure how to incremnt N by that.. Any idea how I can solve this problem? not really sure..

 

[QED]

Originally posted by: jai6638
Hey there..

I have to find the limit of the sequence:

Sqrt(2), Sqrt(2*Sqrt(2)), Sqrt(2*Sqrt(2*Sqrt(2))),....

For an expression, I came up with

Sqrt(2n) but that doesnt work unless, n=1, n=Sqrt(2), n=Sqrt(sqrt2))...

However, not sure how to incremnt N by that.. Any idea how I can solve this problem? not really sure..

Are you still stuck on this?

This is a recursive sequence. If you label the terms in the sequence S_1, S_2, S_3... etc, you will notice that S_n = Sqrt( 2 * S_(n-1) ) for n > 2. In other words, the next term in the sequence is a function of the current term in the sequence.

Very generally speaking, IF IF IF a recursive sequence of the form
S_n = f( S_(n-1) ) actually has a limit (say L), then it will satisfy the equation f(L) = L.

In this case, f(x) = sqrt(2 * x).. so solve x=sqrt(2 * x) and see if that fits. If there is more than one solution to the equation, then there may be more than one value the sequence can converge to depending upon what the initial value of the sequence was.