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Re: Question about yet another bug in Sum
*To*: mathgroup at smc.vnet.net
*Subject*: [mg35222] Re: [mg35204] Question about yet another bug in Sum
*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>
*Date*: Wed, 3 Jul 2002 05:13:20 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
I am not sure what exactly you mean by "the origin of the bug" (writing
code too late at night?) but I can tell you for sure that it is do do
with Limit (as usual, one might add). How do I know this? Well:
<< Calculus`Limit`
Now:
In[2]:=
Sum[Exp[(-I)*Pi*n]*(n/(1 + n^2)), {n, 0, Infinity}]
Out[2]=
(1/4)*PolyGamma[0, 1/2 - I/2] + (1/4)*PolyGamma[0, 1/2 + I/2] -
(1/4)*PolyGamma[0, 1 - I/2] - (1/4)*PolyGamma[0, 1 + I/2]
A nice closed for answer. We can check it agrees numerically with yours:
In[3]:=
N[%]//Chop
Out[3]=
-0.269611
In fact we can check it symbolically:
In[4]:=
Unevaluated[(1/4)*PolyGamma[0,1/2-I/2]+(1/4)*PolyGamma[0,1/2+I/2]-(1/4)*
PolyGamma[0,1-I/2]-(1/4)*PolyGamma[0,1+I/2]==-1/2*
HypergeometricPFQ[{2,1+I,1-I},{2-I,2+I},-1]]//FullSimplify
Out[4]=
True
So life is not so bad after all.
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
On Tuesday, July 2, 2002, at 03:11 PM, Vladimir Bondarenko wrote:
> Hello.
>
> Mathematica is a great environment. I observe that every time before it
> becomes stronger, Mathematica expeciences a collapse ;-)
> Now we at this leg of its majestic trajectory...
>
> Here is another bug in Sum.
>
> $Version
>
> "4.2 for Microsoft Windows (February 28, 2002)"
>
> Sum[Exp[-I Pi n] n /(1 + n^2), {n, 0, Infinity}]
>
> ACTUAL: Sum::"div": "Sum does not converge."
>
> EXPECTED: -1/2*HypergeometricPFQ[{2, 1 + I, 1 - I}, {2 - I, 2 + I},
> -1]
>
> CHECK-UP: N[%, 20]//Chop
>
> -0.26961050270800898180
>
> NSum[Exp[-I Pi n] n/(1 + n^2), {n, 0, Infinity},
> WorkingPrecision->20]//Chop
>
> -0.269610502708010
>
> This bug is present in 4.1, 4.0, 3.0. Version 2.2 leaves the sum
> unevaluated.
>
>
> Can anyone explain the origin of the bug?
>
>
> Best,
>
> Vladimir Bondarenko
>
>
>
>
>
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