Borel summation

[Gottfried]

But with this approach we might not need Borel summation at all, we can just take a truncation of the divergent series - say \(h_N(x)\) - and set (...)

Well, something like this has been the reason for my no-continuing with the analysis/application Borel-summation at all: practically not better than my handy matrix-summation method, ready made for use in Pari/GP.        

But I always kept the reserve, that one were assuming the sufficience of the Borel-method, and was only awaiting the working-out of a proof-of-concept (which we seem to have now by James and his literature-dig).
Thus no sophisticated elaborations with tables & plots from my side...

Gottfried