I also posted a similar question on Stack Exchange.

The MRB constant is one of many that you may find in the Mathematica resources. Enter the double equals and the name of the MRB constant to see that. You can also do math using that double equals. Now, the MRB constant is the limit point defined by $ \underset{x\to \infty }{\text{lim}}\left(\sum _{n=1}^{2 x} (-1)^n n^{1/n}\right)$ This sequence involves the nth roots of n, which can be interpreted geometrically as when across all hypercubes with hyper volume equal to n, the alternating sum of these edge lengths converges to the MRB constant.

Since a unit hypercube's longest diagonal in n dimensions is equal to $\displaystyle {\sqrt {n}}$, and the diameter of an n-dimensional hypercube is given by the longest possible diagonal, which is n times the square root of the length of one of its edges, if we define the edge length as $ s = n^{1/n} $, then substituting: $ d = \sqrt{(n^{2/n} \times n^{n/n})} = \sqrt{n^2} = n $

If the above is correct, this seems to suggest that the diameter equals ( n ), which links it to the MRB constant and dimensional scaling. Thus, across all hypercubes with diameter ( n ), the alternating sum of these edge lengths converges to the MRB constant.

My derivation of that statement seems too half-hazardous. How can I formalize it?

I'm having trouble getting the Wolfram Notebook Assistant to help me here, so could I be missing something in my explanation?