Let $0<a\neq 1$ be a fixed real number and denote by $a^{ \frac{x}{}}:=\mbox{uxp}_a(x)$ the ultra exponential function (ultra power) that is a unique extension of tetration (and also its linear approximation). One can see the uniqueness theorem in https://en.wikipedia.org/wiki/Tetration and the paper :Ultra power and ultra exponential functions (http://www.tandfonline.com/doi/abs/10.1080/10652460500422247).
Note that $$ \mbox{uxp}_a(x)=\exp^{[x]}_a (a^{(x)})=\exp^{[x+1]}_a((x))= \begin{cases} \vdots \;\;\;\;\; & \; \vdots\\ \log_a(x+2) \;\;\;\;\; & \; -2<x<-1 \\ 1+x \;\;\;\;\; & \; -1\leq x<0 \\ a^x \;\;\;\;\; & \; 0\leq x<1 \\ a^{a^{x-1}} \;\;\;\;\; & \; 1\leq x<2 \\ \vdots \;\;\;\;\; & \; \vdots \end{cases}$$ where $[x]$ is the largest integer not exceeding $x$, $(x)=x-[x]$ (the fractional part of $x$), $\exp_a (x)=a^x$, $\exp(x)=\exp_e(x)=e^x$.
(?) Now, is it true that the following series is convergent for $a=(\frac{1}{e})^e$ and all $x>-2$ ? $$ \sum_{n=1}^{\infty}a^{\frac{n}{}}-a^{\frac{n+x}{}} $$
