A fairly standard problem in the study of sequences is to make rigorous sense of the expression
$$
\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}
$$
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.