I've been looking into two very specific forms that are related to $\sum _{n=1}^{\infty } (-1)^n \left(n^{\frac{1}{n}}-1\right)$ cf http://mathworld.wolfram.com/MRBConstant.html.
They are
$\sum _{n=1}^{\infty } (-1)^n \left(n^{\frac{u}{n}}-1\right)$ and $\sum _{n=1}^{\infty } (-1)^n \left(n^{\frac{u}{n}}-u\right)$.
As the amateur that I am, I would like to ask academic (or other) professionals if they have ever come across these forms, or are they clearly localized? To see where I get these two forms you can cf http://www.mapleprimes.com/posts/104016-A-Very-False-Graph.
