While browsing at Barnes & Noble, I came across Excursions in Number Theory by Ogilvy and Anderson. “Hey! I teach number theory,” I thought. So I flipped through the book for a while and came across the following question:
Let be an integer, and consider the expression
Does this ever converge? If so, to what?
Assume the expression converges (what it means for something like that to converge is left for another post), and let be the limit. Then
and . In particular, the expression converges. What’s more, if for some integer , then
an integer! For example, with , so that , we have
Neat! Mathworld has a lot more on nested radicals. Especially cool are the expressions for :
and the golden ration :