08/10/2022, 02:12 PM
Yeah Gotfried at first sight about the a_k terms I thought about Borel summation, and I was also interested into divergent sums
I once also stucked in the summing-up terms d_n=O(exp(exp(n))) which is very much faster growing than O(n!) and any O(n!^k), I studied O(exp(exp(n))) because of the infinity q-pochhammer function \((q;q)_\infty\) which has nice expansions but O(q^n^2) cannot sum and even analytically extended to \(norm(q)>1\). I was so convinced about the contour integral method could sum diverge summations up, this may help idk
This may belong to the barren land of modern maths, but still I think it can guarantee a finite value as the sum, for example I used to extend the function
\(f(z)=\sum_{n\ge0}{z^{2^n}}\), firstly I used contour integration and it fits well, and it did also work outside \(\norm(q)\le1\), and also I used the property \(f(z^2)=f(z)-z\) to discover it's multivalued, and it might be a success?
I once also stucked in the summing-up terms d_n=O(exp(exp(n))) which is very much faster growing than O(n!) and any O(n!^k), I studied O(exp(exp(n))) because of the infinity q-pochhammer function \((q;q)_\infty\) which has nice expansions but O(q^n^2) cannot sum and even analytically extended to \(norm(q)>1\). I was so convinced about the contour integral method could sum diverge summations up, this may help idk
This may belong to the barren land of modern maths, but still I think it can guarantee a finite value as the sum, for example I used to extend the function
\(f(z)=\sum_{n\ge0}{z^{2^n}}\), firstly I used contour integration and it fits well, and it did also work outside \(\norm(q)\le1\), and also I used the property \(f(z^2)=f(z)-z\) to discover it's multivalued, and it might be a success?
Regards, Leo 
