Let $a,b\in\Bbb{N}$. Is there a closed form for $\displaystyle\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$ ? (I use Knuth's up arrow notation)
If so, how can we obtain it ?
If there isn't a closed form, is there at least a 'nicer' way to express it using other known functions ?
This is not homework, I'm just asking out of curiosity.