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NSum[(-1)^n*(n^(1/n)-a),{n,Infinity}] and the like

  • To: mathgroup at smc.vnet.net
  • Subject: [mg120038] NSum[(-1)^n*(n^(1/n)-a),{n,Infinity}] and the like
  • From: Marvin Burns <marvin at marvinrayburns.com>
  • Date: Tue, 5 Mar 2013 22:16:41 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • Delivered-to: l-mathgroup@wolfram.com
  • Delivered-to: mathgroup-newout@smc.vnet.net
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Ideally I would like to see someone explain why I get the following results 
they might also be of interest to others.
For all a, x, and y Mathematica gives the following where c is the constant 
given by the convergent series c=NSum[(-1)^n*(n^(1/n)-1),{n,Infinity}] 
= 0.18785964246206... = the MRB constant.
Regularization is used so sums that formally diverge return a result that can be 
interpreted as evaluation of the analytic extension of the series:
NSum[(-1)^n*(n^(1/n)-a),{n,Infinity}] gives c-1/2*(1-a).
NSum[(-1)^n*(x*n^(1/n)+y*n),{n,Infinity}] gives (c-1/2)*x-1/4*y.
NSum[(-1)^n*(x*n^(1/n)-a),{n,Infinity}] gives (c - 1/2)*x + 1/2*a.



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