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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.)
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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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CAS: sometimes doesn't detect Zeta series (Bug?) - danielmewes - 01-04-2015 08:27 AM

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