Fall 2016 Challenge Problem #1

A fairly standard problem in the study of sequences is to make rigorous sense of the expression
and find its value. In this instance, one looks at the sequence of “partial” expressions obtained by cutting the given expression off after only finitely many radicals and $2$s. It is easy to see that the sequence of these is increasing and bounded above by $2$, so must converge to a limit $L$. It is straightforward that then $L$ must satisfy the equation $L = \sqrt{2+L}$, from which it $L = 2$.

There have been variations of this problem among our past challenge problems, and our current offering is another one.

Problem. Make rigorous sense of the expression
L = \sqrt{1 – \sqrt{2- \sqrt{3-\sqrt{4-\sqrt{5-\cdots}}}}}
and determine whether $L$ has a numerical value. If it does, you are not required to evaluate it.

Remark. Real numbers only!

Solutions are due by email by 3:00 p.m. on Thursday 22 September 2016.