User blog comment:LittlePeng9/Fast growing hierarchy of analytic functions/@comment-30754445-20170617192307/@comment-1605058-20170618122527

The only tip I can think of is that for $$|z|<2^n$$ you can stop the summation after $$n$$ terms, since the remaining summands add up to at most $$1$$.

I have devised these functions without a practical purpose in mind, so I didn't think of making them easy to compute. I may think of a way to make that easier, but I doubt we can really do without using an infinite sum or a product, which is probably the main problem in here.