Re: [Maxima-discuss] simplify_sum leads to a wrong value

[Stern, M. (NIH/NIA/I. [E] <st...@ma...>]

I tried philippe's examples on the closedform function in Macsyma 2.4 (due to Gosper) and it couldn't do either of them.  Then I experimented with one that Macsyma can do:

sum(k^2/(k^4+1),k,1,inf)

Macsyma gives a result in terms of complex trig functions which simplifies to 1.125...
Maxima simplify_sum gives a result in terms of complex psi[0] which neither maxima nor Macsyma can simplify or even prove to be real.  Perhaps there is just a need for more knowledge about psi functions.

-----Original Message-----
From: Raymond Toy <toy...@gm...> 
Sent: Wednesday, April 8, 2020 11:04 AM
To: max...@li...
Subject: Re: [Maxima-discuss] simplify_sum leads to a wrong value

>>>>> "philippe" == philippe  <rou...@gm...> writes:

    philippe> Hi,
    philippe> I've check the results with SAgeMath (which obviously call maxima for
    philippe> this computation) and wolframalpha. This last one found another closed
    philippe> form for the series :

    philippe> https://photos.app.goo.gl/XGoyMm9TCa6t34gs7

Interesting.  Maxima can simplify the 2F1 function:

hgfred([1/2,2],[5/2],x);
(sqrt(x)*(3*atanh(sqrt(x))*x+3*atanh(sqrt(x)))-3*x)/(4*x^2)$

If you evaluate this at x = -(-1)^(2/3) and x = (-1)^(1/3) you get 3/4 for both.  Which is pretty obvious since (-1)^(1/3) = -1 and
-(-1)^(2/3) = -1, assuming principal roots.  This gives

((-1)^(1/6)-(-1)^(5/6))/8 = 0.2165063509461097

Which matches neither your expected result nor maxima's simplify_sum result.

It's not clear which of the three roots should be used in the formula.  

Wolfram Alpha gives the same simplified result that hgfred does, so at least that's not to blame.




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