Since the MRB constant is an alternating sum of positive integers to their own roots, f(n)=(-1)^n* n^(1/n); a thorough understanding of the changes in f, as n changes, is important.
In this blog we will begin to explore the derivative of f at integer values of n, and as n-> infinity. I am not sure weather this will help us in computing more digits of the MRB constant since we already know so many, 250,000; rather, it is my hope that the study of the derivative of f will help in finding the illusive closed form for Limit(Sum((-1)^n*n^(1/n), n = 1 .. 2*N), N = infinity), if such a form does indeed exist.
Notice that if a function is differentiable at some point c of its domain, then it is also continuous at c. However here we extend the notion of differentiability to be valid for individual points on the real number line, specifically positive integers.
