A mathematical problem

I submitted the problem outlined here to the American Mathematical Monthly and they seem willing to publish it in their monthly problems page. One of the collaborating editors has provided a proof of part (b), which is delightful news. It takes a route I considered briefly but then discarded as being impassable!

The only thing missing in the editor’s proof is that the error in the approximation is proved to be o(1) , whereas I really need O(1/n) and a good idea of what multiple of 1/n is involved. Perhaps one of the solvers will improve the error term once the problem appears in the magazine.