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Science Forum Index » Math - Symbolic Forum » Problem with a hypergeometric summation in Mathematica...
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| Fredrik... |
 Posted: Sat Aug 02, 2008 9:53 pm |
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Mathematica 6.0 for Linux x86 (32-bit)
Copyright 1988-2008 Wolfram Research, Inc.
In[1]:= Sum[n/(n^3+1), {n, 1, Infinity}]
1 - EulerGamma 1/3
Out[1]= -------------- + ((1 + (-1) )
3
1/3 2/3
Quote: HypergeometricPFQ[{2, (-1) , -(-1) },
1/3 2/3
Quote: {1 + (-1) , 1 - (-1) }, 1] +
2/3 1/3 2/3
Quote: (-1) (-PolyGamma[0, (-1) ] + PolyGamma[0, -(-1) ])) /
1/3
In[2]:= N[%]
Out[2]= ComplexInfinity
But the sum is obviously finite. Is the problem with Sum or N? In
either case, is there a workaround (short of using NSum)?
Fredrik |
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| Raymond Manzoni... |
| Posted: Sun Aug 03, 2008 3:21 am |
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Fredrik a écrit :
Quote: Mathematica 6.0 for Linux x86 (32-bit)
Copyright 1988-2008 Wolfram Research, Inc.
In[1]:= Sum[n/(n^3+1), {n, 1, Infinity}]
1 - EulerGamma 1/3
Out[1]= -------------- + ((1 + (-1) )
3
1/3 2/3
HypergeometricPFQ[{2, (-1) , -(-1) },
1/3 2/3
{1 + (-1) , 1 - (-1) }, 1] +
2/3 1/3 2/3
(-1) (-PolyGamma[0, (-1) ] + PolyGamma[0, -(-1) ])) /
1/3
(3 (1 + (-1) ))
In[2]:= N[%]
Out[2]= ComplexInfinity
But the sum is obviously finite. Is the problem with Sum or N? In
either case, is there a workaround (short of using NSum)?
Fredrik
I don't know for Mathematica but MuPAD v3 returned :
sum(n/(n^3+1),n=1..infinity)
1/2 1/2
psi(1/2 - 1/2 I 3 ) (I 3 - 1) EULER
---------------------------------- - ----- -
6 3
1/2 1/2
psi(1/2 I 3 + 1/2) (I 3 + 1)
---------------------------------- + 1/3
6 |
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| Raymond Manzoni... |
| Posted: Sun Aug 03, 2008 3:33 am |
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Raymond Manzoni a écrit :
Quote:
I don't know for Mathematica but MuPAD v3 returned :
sum(n/(n^3+1),n=1..infinity)
1/2 1/2
psi(1/2 - 1/2 I 3 ) (I 3 - 1) EULER
---------------------------------- - ----- -
6 3
1/2 1/2
psi(1/2 I 3 + 1/2) (I 3 + 1)
---------------------------------- + 1/3
6
The numerical value of this is nearly 1.11164393822406629494121491215
The expression may be obtained using partial fraction on n/(n^3+1)
and the classical formulas for psi :
http://www.convertit.com/Go/Bioresearchonline/Reference/AMS55.ASP?Res=150&Page=265&Submit=Go
Hoping it helped anyway...
Raymond |
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| Axel Vogt... |
| Posted: Sun Aug 03, 2008 4:50 am |
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Fredrik wrote:
Quote: Mathematica 6.0 for Linux x86 (32-bit)
Copyright 1988-2008 Wolfram Research, Inc.
In[1]:= Sum[n/(n^3+1), {n, 1, Infinity}]
1 - EulerGamma 1/3
Out[1]= -------------- + ((1 + (-1) )
3
1/3 2/3
HypergeometricPFQ[{2, (-1) , -(-1) },
1/3 2/3
{1 + (-1) , 1 - (-1) }, 1] +
2/3 1/3 2/3
(-1) (-PolyGamma[0, (-1) ] + PolyGamma[0, -(-1) ])) /
1/3
(3 (1 + (-1) ))
In[2]:= N[%]
Out[2]= ComplexInfinity
But the sum is obviously finite. Is the problem with Sum or N? In
either case, is there a workaround (short of using NSum)?
Fredrik
Maple 12 gives it as
1/3-1/3*Psi(1/2+1/2*I*3^(1/2))
+1/6*I*Pi*tanh(1/2*Pi*3^(1/2))
+1/6*3^(1/2)*Pi*tanh(1/2*Pi*3^(1/2))
-1/3*gamma, gamma=Euler's constant |
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| David W. Cantrell... |
| Posted: Sun Aug 03, 2008 6:22 am |
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Fredrik <fredrik.johansson at (no spam) gmail.com> wrote:
Quote: Mathematica 6.0 for Linux x86 (32-bit)
Copyright 1988-2008 Wolfram Research, Inc.
In[1]:= Sum[n/(n^3+1), {n, 1, Infinity}]
1 - EulerGamma 1/3
Out[1]= -------------- + ((1 + (-1) )
3
1/3 2/3
HypergeometricPFQ[{2, (-1) , -(-1) },
1/3 2/3
{1 + (-1) , 1 - (-1) }, 1] +
2/3 1/3 2/3
(-1) (-PolyGamma[0, (-1) ] + PolyGamma[0, -(-1) ])) /
1/3
(3 (1 + (-1) ))
In[2]:= N[%]
Out[2]= ComplexInfinity
But the sum is obviously finite. Is the problem with Sum or N? In
either case, is there a workaround (short of using NSum)?
Raymond has already given you a mathematical answer, but here's
an easy Mathematica workaround:
In[3]:= Sum[n/(n^3 + a), {n, 1, Infinity}]
Out[3]= (-(1/3))*RootSum[1 + a + 3*#1 + 3*#1^2 + #1^3 & ,
PolyGamma[0, -#1]/(1 + #1) & ]
In[4]:= % /. a -> 1
Out[4]= (-(1/3))*RootSum[2 + 3*#1 + 3*#1^2 + #1^3 & ,
PolyGamma[0, -#1]/(1 + #1) & ]
In[5]:= N[%]
Out[5]= 1.111643938224066 + 0.*I
I suggest that you report your bug to Wolfram Research.
David |
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