If we want. We can try to treat this as a uniqueness claim for extending hyperoperators to complex numbers. That if we find a formula for that satisfies recursion for natural numbers it should also satisfy recursion for complex numbers. Ergo; uniqueness for how hyper operators behave for complex numbers. It's not very lucid yet. Again; one step at a time.

EDIT:

evaluation of limit

Therefore we can rephrase the limit as; keeping in mind the first term disappears:

Which means convergence is inconclusive. Using this method generally we will have this result; so long as remains a quotient. Anyone know any other convergence tests? I'm pretty sure this outlaws the root test as inconclusive as well. The integral test? I'll keep looking... Maybe I made a mistake. I tried using the integral test and I got divergence. The methods were a little questionable however. I'll hold out on posting that just yet.

EDIT:

evaluation of limit

Therefore we can rephrase the limit as; keeping in mind the first term disappears:

Which means convergence is inconclusive. Using this method generally we will have this result; so long as remains a quotient. Anyone know any other convergence tests? I'm pretty sure this outlaws the root test as inconclusive as well. The integral test? I'll keep looking... Maybe I made a mistake. I tried using the integral test and I got divergence. The methods were a little questionable however. I'll hold out on posting that just yet.