bo198214 Wrote:This is ingenious!
Quote:So we have for each coefficient two sequences, where for each the n+1-th element can be computed only from the previous n elements, and the coefficient is the scalar product of both sequences.
Perhaps we can solve the convergence problem with this decomposition.
Yes, that would be good!
I'm tinkering a bit with the problem (but not yet seriously): why is that series so misconfigured for convergence-acceleration with Euler/Cesaro-sum? Is there some quantity, which far-away coefficients "have to compensate" so they cannot decrease to zero (something similar to a carry in the core addition- or multiplication-procedures in a processor)?
We see, that the constant value for the series is -1, which is the expected value for the slog at x=0, and which gives a good rationale for this setting.
But what, if we could rearrange the problem around the initial condition, that we had a constant value of -2, for log_b(0)? Perhaps we would get a series which behaves nicer...
Well, I observe, that this alludes to the "nice serie"-msgs in the other thread, where we indeed deal with the "-2" -constant.
Hmm... I even don't have an idea how to access such a change...
Well, let's first carry home the fruits from the current results. Any further improvement can be attempted later.
Gottfried Helms, Kassel