Nested RadicalDate: 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. |
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