Prove that sqrt(1+sqrt(1+sqrt(1+sqrt(...)))) = the golden ratio (ILIEKMATHPHYSICS)

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This video references the book "Introduction to Real Analysis" by Bartle and Sherbert (Fourth Edition). For more details, see section 3.3 on the Monotone Convergence Theorem.

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Автор

I just love these! every singe step clearly explained! :))

vlynn
Автор

To prove the fact that it's increasing, use proof by induction...

Base case: x1 = 1, x2 = sqrt(2), so x2 > x1. Easy base case.

Inductive case: Assume x(n) > x(n-1). Then 1 + x(n) > 1 + x(n-1). Therefore, sqrt(1 + x(n)) > sqrt(1 + x(n-1)). Therefore x(n+1) > x(n). Inductive case done.

chaosredefined