(12/14/2009, 01:33 PM)bo198214 Wrote:(12/08/2009, 10:32 AM)mike3 Wrote: So far, I've found 2 types of transseries representation such that if the series converges, the continuum sum does as well (proof given earlier here):

I was a bit puzzled why converges via transseries while does not converge. Actually must also converge as we have , and indeed it also converges! But it seems that this already the limit, i.e. every series for does not converge. I guess that this is the same for , , while Faulhaber always works for , i.e. . Do you have any idea how to sum ?

Hi,

I've been reading on this forum for a while, and just now I've signed up.

I have been exploring the sum operator, what you guys call "continuum sum" for a couple of years as a hobby.

I think I have an idea how to sum .

Recall that

Now calculate the polynomial sum of (using Faulhaber's formula).

This seems to convergent to a nice function.