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)
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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.