Expert: Jack Cheng - 1/5/2007

Question
hi
im having a little trouble finding the limit of this sequence:
root(2), root(2*root(2)), root(2*root(2*root(2))), root(2*root(2*root(2*root(2))))
help would be greatly appreciated!
thanks

Answer
Hi Ammar,

Each number gives you one hint, and each hints gives you more information to help you solve the problem. Stop at each hint and see if that's enough to help you. Don't go to the last hint immediately. You don't want to spoil the fun in solving this problem now, right?


1) Can you rewrite the sequence in a manner that is easier to work with, or conceptualize? If you can do that, the problem is almost solved.
























2) root(x*y) = root(x)*root(y)

















3) root(x) = x^(1/2), so root(root(x)) = (x^(1/2))^(1/2) = x^(1/4)

Can you rewrite the sequence now?






















4) The sequence can be rewritten as 2^(1/2) * 2^(1/4) * 2^(1/8) * ... = 2^(1/2 + 1/4 + 1/8 + ...)





















5) The series 1/2 + 1/4 + 1/8 + ... is equaled to 1.























6) The answer is 2.



I hope you like this problem,
Jack