CAS: sometimes doesn't detect Zeta series (Bug?)

01042015, 08:27 AM
(This post was last modified: 01042015 08:32 AM by danielmewes.)
Post: #1




CAS: sometimes doesn't detect Zeta series (Bug?)
Per definition, `sum(1/n^s, n, 1, inf) = Zeta(s)` for s > 1.
However Giac doesn't detect this fact consistently, instead failing on evaluating the sum. These work:
These however don't work:
I would wish that all of these examples would evaluate to `Zeta(s)` (or the exact evaluation thereof we possible, like for even integers `s`). In addition, the freeze and apparently nondeterministic behavior of `sum(1/n^(5/3), n, 1, inf)` suggest a second bug on the Prime. 

01052015, 02:54 PM
Post: #2




RE: CAS: sometimes doesn't detect Zeta series (Bug?)
There is indeed a bug in the normal code, called by the hypergeometric recognition function, and also a bug in this one. They are now fixed in source and in Xcas.
I do not plan to add recognition of sum(1/n^k,n,...,inf) to Zeta(k) now. Unlike for rational fractions where there is an algorithm (partial fraction expansion), it would only work for this serie, it's too much work for almost no benefit (especially since there is no closed form of Zeta for rationals in giac). 

01062015, 03:47 AM
Post: #3




RE: CAS: sometimes doesn't detect Zeta series (Bug?)
Thanks Bernard!
Quote:I do not plan to add recognition of sum(1/n^k,n,...,inf) to Zeta(k) now. Unlike for rational fractions where there is an algorithm (partial fraction expansion), it would only work for this serie, it's too much work for almost no benefit (especially since there is no closed form of Zeta for rationals in giac). Sure, sounds reasonable. The only benefit I can see in transforming this to Zeta(k) is that it would allow using the numeric approximation of Zeta (from GSL?) for such sums. I understand that it's not worth a lot of work though. 

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