(08/21/2022, 08:54 AM)bo198214 Wrote: Actually I wonder why Gottfried didnt post any results about the divergent summations he tried.
upps, a little introduction into this is in this exposition .
The previous-to-last table shows Noerlund-summantion (q&d-homebrewn) with 64 coefficients of the halfiterate of exp(x)-1 - where the partial sums look good -, but increasing the number of coefficients later showed a new beginning of divergence in that partial sums.
After that, in the last table I show this with 256 coefficients and a stronger parameter for the Noerlund-sum, and here I seem to have got it.
Why not shown more results here?
This implementation of the divergent-summation-scheme is somehow similar to the simple Borel-summation as shown in K.Knopp's book (chap 13, german edition), so I've been confident that I -at least principally- knew that I can apply that method. (Note, that I did not yet have this idea for the limiting function as shown here in the current threads, this came only later 4 years or so). On the other hand I didn't see the need to document my numerical results here, since after the successful use of asymptotic series here around and even more after Sheldon's routines, there seemed to be no more need for concurring methods for actual numerical computation with never really perfect (satisfying) approximations...
In the view back it is interesting that I didn't have the idea of 4 subsequences in the coefficients, which would have made my last picture more meaningful; but the pattern there was already so remarkable and unique that I applied this representation to other series as well - trying to discern more precisely what's going on there at all - but, well, after the idea of the limiting function I think I can omit know that pictures/representations...
Gottfried Helms, Kassel
