What is the limit to this sequence?

√(2)
√(2+√(2))
√(2+√(2+√(2)))
√(2+√(2+√(2+√(2))))...

In general, what is the sequence in terms of x if 2 is replaced with x?

If we call S the 'n' term of this sequence, like:
S = √(2+√(2+√(2+...
and we square both sides of the equation, we will reach:
S² = 2+S
S²-S-2=0
S=(1(+/-)√(1+2*4))/2, and considering only the positive solution as valid for our purpose, we reach S=2 as being the limit to this sequence.
And if 2 is replaced with X, the solution would be S=((1+√(1+4*X))/2 (again considering only the positive solution as valid for us).

Posted by Roberto Mattos on 2004-02-20 12:29:14