       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:=
Sum[Exp[(-I)*Pi*n]*(n/(1 + n^2)), {n, 0, Infinity}]

Out=
(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:=
N[%]//Chop

Out=
-0.269611

In fact we can check it symbolically:

In:=
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=
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,
>