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Hi All

Been investigating lately ways to sum ordinarily divergent series. Looked into Cesaro and Abel summation, but since if a series is Abel Mable it is also Cesaro sumable, but no, conversely,haven't worried about Cesaro Summation. Noticed Abel summation is really a regularization technique similar to regularization in re-normalization then you let x in the regulator x^n go to 1.

It all looked good until you looked art series like 1+1+1+1 ... or 1+2+3+4 etc ie the terms are not 'oscillating' like say 1-2+3-4 etc - then it fails. Of course zeta function summation works and is related to re-normalization as worked out by Hawking:

https://projecteuclid.org/euclid.cmp/1103900982

There are hand-wavy ways to use Abel summation to handle 1+2+3... - but it's a bit iffy IMHO as explained in a heap of places on the internet eg (the following has been, correctly criticized a lot))

But even if you accept it, it fails miserably for 1+1+1+1...

Ok you can use analytic continuation on the zeta function - but how did Ramanujan do it? I started looking into that. Amazingly I found a good video on it:

The answer is easy actually - its the constant term in the Euler-MacLaurin summation formula. Well I will be stonkered - its that easy.

If you would like the full detail see (note to other moderators is its a copy the author make freely available of a textbook he wrote on it so meets our standards for a reference):

https://hal.univ-cotedazur.fr/hal-01150208v2/document

Thanks

Bill

Been investigating lately ways to sum ordinarily divergent series. Looked into Cesaro and Abel summation, but since if a series is Abel Mable it is also Cesaro sumable, but no, conversely,haven't worried about Cesaro Summation. Noticed Abel summation is really a regularization technique similar to regularization in re-normalization then you let x in the regulator x^n go to 1.

It all looked good until you looked art series like 1+1+1+1 ... or 1+2+3+4 etc ie the terms are not 'oscillating' like say 1-2+3-4 etc - then it fails. Of course zeta function summation works and is related to re-normalization as worked out by Hawking:

https://projecteuclid.org/euclid.cmp/1103900982

There are hand-wavy ways to use Abel summation to handle 1+2+3... - but it's a bit iffy IMHO as explained in a heap of places on the internet eg (the following has been, correctly criticized a lot))

But even if you accept it, it fails miserably for 1+1+1+1...

Ok you can use analytic continuation on the zeta function - but how did Ramanujan do it? I started looking into that. Amazingly I found a good video on it:

The answer is easy actually - its the constant term in the Euler-MacLaurin summation formula. Well I will be stonkered - its that easy.

If you would like the full detail see (note to other moderators is its a copy the author make freely available of a textbook he wrote on it so meets our standards for a reference):

https://hal.univ-cotedazur.fr/hal-01150208v2/document

Thanks

Bill

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