CAS: sometimes doesn't detect Zeta series (Bug?)
01-04-2015, 08:27 AM (This post was last modified: 01-04-2015 08:32 AM by danielmewes.)
Post: #1
 danielmewes Junior Member Posts: 29 Joined: Dec 2014
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:
• sum(1/n^2, n, 1, inf) evaluates to 1/6 * pi^2 (i.e. Zeta(2))
• sum(1/n^3, n, 1, inf) evaluates to Zeta(3), which can be numerically evaluated through approx(Zeta(3))
• in fact it seems that for any integer s>1 sum(1/n^s, n, 1, inf) evaluates correctly

These however don't work:
• sum(1/n^2.5, n, 1, inf) "evaluates" to sum(1/n^2.5, n, 1, inf)
• same for any other approximate number s
• sum(1/n^(5/2), n, 1, inf) prints four warnings of the form "Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [<varying number. Different in each run, for example: 10, 79.52..., 4, 82.899...>]". Then just returns the original expression slightly rewritten sum(1/(sqrt n * n^2), n, 1, inf).
• sum(1/n^(5/3), n, 1, inf) occasionally freezes the Prime (not interruptible by pressing On, nor On+Symb. Requires hard reset). Other times it returns a big useless expression that still contains n, but no summation symbol or other definition of it: -ln(abs(-(n^(1/3))²/((n+1)^(1/3))²+1))*((n+1)^(1/3))²/(n^(1/3))². On the PC with Xcas (Giac 1.1.3), this always evaluates to the slightly rewritten input sum(1/((n^(1/3))^2*n), n, 1, inf)
• other fractions for s don't seem to work either, unless they simplify to an integer (e.g. sum(1/n^(4/2), n, 1, inf) works, since 4/2 = 2).

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 non-deterministic behavior of sum(1/n^(5/3), n, 1, inf) suggest a second bug on the Prime.
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 Messages In This Thread CAS: sometimes doesn't detect Zeta series (Bug?) - danielmewes - 01-04-2015 08:27 AM RE: CAS: sometimes doesn't detect Zeta series (Bug?) - parisse - 01-05-2015, 02:54 PM RE: CAS: sometimes doesn't detect Zeta series (Bug?) - danielmewes - 01-06-2015, 03:47 AM

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