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Alladi-GrinsteadConstant

  • To: mathgroup at smc.vnet.net
  • Subject: [mg113189] Alladi-GrinsteadConstant
  • From: Peter Pein <petsie at dordos.net>
  • Date: Sun, 17 Oct 2010 06:06:46 -0400 (EDT)

Dear Group,

please have a look at the following:

It is OK for me to return an unevaluated sum:
In[1]:= Sum[-(Log[1 - 1/k]/k), 
  {k, 2, Infinity}]
Out[1]= Sum[-(Log[1 - 1/k]/k), 
  {k, 2, Infinity}]

it has got a value:

In[2]:= NSum[-(Log[1 - 1/k]/k), 
  {k, 2, Infinity}]
Out[2]= 0.7885305658471413

to try it as double sum, get the seriescoeffs:

SeriesCoefficient[-(Log[1 - 1/k]/k), 
  {k, Infinity, n}]
Out[3]= Piecewise[{{1/(-1 + n), n >= 2}}, 0]

and feed this into Mathematica:

Sum[1/(k^n*(n - 1)), {n, 2, Infinity}, 
  {k, 2, Infinity}]
Out[4]= -Infinity

MINUS Infinity????
The same with a big N:

NSum[1/(k^n*(n - 1)), 
  {n, 2, Infinity}, {k, 2, Infinity}]
Out[5]= 0.7885305658584437
In[6]:= Exp[% - 1]
Out[6]= 0.8093940204976885
In[7]:= $Version
Out[7]= 7.0 for Linux x86 (64-bit) (February 18, 2009)

Will this be fixed in Version 8?



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