Expert: Paul Klarreich - 4/4/2008

Question
Hi Paul,
How do I find the limit of a sequence described by
sqrt(2), sqrt(2(sqrt(2)), sqrt(2sqrt(2(sqrt(2)))...
Thanks!

Answer
Questioner:   vince
Category:  Calculus
Private:  No
 
Subject:  sequence /limits
Question:  Hi Paul,
How do I find the limit of a sequence described by
sqrt(2), sqrt(2(sqrt(2)), sqrt(2sqrt(2(sqrt(2)))...
Thanks!
..................................
Hi, Vince,

This is a kind of general method you might try for stuff like this.

ASSUMING the sequence converges, let  x be the limiting term.  Then there really is no difference between consecutive terms.  So we can write:

x = sqrt(2x)

and solve it.

x^2 = 2x

x = 2 or x = 0.

Clearly x = 0 is not the solution, but x = 2 might be.

A quick-and-dirty calculation (I use Excel) confirms this.