Timeline for A proposition for summing divergent series, but how should partial summation be defined at non-natural values?
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| Oct 15, 2021 at 17:28 | comment | added | Caleb Briggs | If I'm not mistaken, the method used in that paper of instead looking at $\lim_{n\to \infty} f(-n)$ roughly corresponds to recentering the power series at $x=\infty$, at least in the case of $\sum_{n=0}^\infty f(-n)x^{n} \mapsto -\sum_{n=1}^\infty f(-n)x^{-n}$. I haven't rigorously looked at it, but I think the left summation will fail to converge in places between singularities, for instance, if there is a singularity at some point when |z|=1 and |z|=2, then when 1<|z|<2 it seems to fail. I suppose it should fail at some points when $f(n)$ looks like $f(n) = a^n + (1/b)^{-n}$ | |
| Oct 15, 2021 at 0:38 | comment | added | Jorge Zuniga | @CalebBriggs, It seems that polynomial growth rate condition (Axiom S6 in [3]) can be relaxed (perhaps, deleted) for some fast divergent cases. In fact, such article shows that sums of linearly divergent series $\sum_{n=0}^\infty x^n$ for $x > 1$, whose partial sums grow beyond polynomial growth are properly computed using fractional (left) sums. I will check how far can this be stretched (factorial divergence and beyond) | |
| Oct 14, 2021 at 22:37 | comment | added | Caleb Briggs | This is a helpful and interesting answer (thank you for including lots of sources), however, the fractional summation you mentioned only works for functions that approximately have a polynomial growth rate. I'm interested in finding ways to define fractional summation for much faster growing functions, since all of the functions I investigate grow faster than polynomial speed. | |
| Oct 14, 2021 at 21:08 | history | answered | Jorge Zuniga | CC BY-SA 4.0 |