Nested Radical

Date: 02/25/2002 at 19:58:50
From: Mac
Subject: Nested Radical

Please show me how to prove the following nested radical.

   sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+...)))) = 3

Thank you very much.

Date: 02/25/2002 at 21:01:47
From: Doctor Wilkinson
Subject: Re: Nested Radical

Nice problem, Mac! Here's the solution given by Ramanujan. You'll need 
to fuss a bit about convergence.

Consider the identity

  [x+n]^2 = n^2 + 2nx + x^2 = n^2 + x [(x+n) + n]

Take square roots to get

  [x+n] = sqrt(n^2 + x [(x+n)+n])

Now inside the brackets you have something + n, so you can substitute 
in the same equation with x+n replacing x to get

 x+n = sqrt(n^2 + x sqrt(n^2 + (x+n)[(x+2n)+n]))

and repeat the process to get

 x+n = sqrt(n^2 + x sqrt(n^2 + (x+n) sqrt(n^2 + (x+2n) sqrt(...)...)

If you now set n = 1, x = 2 you get

  3 = sqrt(1 + 2 sqrt(1 + 3 sqrt(1 + 4 sqrt(...)...)

- Doctor Wilkinson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/25/2002 at 22:19:24
From: Mac
Subject: Nested Radical

Thank you very much. I really appreciated your response.