Re: Harmonic Numbers
- To: mathgroup at smc.vnet.net
- Subject: [mg107906] Re: [mg107881] Harmonic Numbers
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Tue, 2 Mar 2010 07:53:54 -0500 (EST)
- Reply-to: hanlonr at cox.net
$Version
7.0 for Mac OS X x86 (64-bit) (February 19, 2009)
s = Sum[k/((k^2 + 1) (k^2 + 4)), {k, 1, Infinity}]
(-(1/2))*RootSum[#1^4 + 4*#1^3 +
11*#1^2 + 14*#1 + 10 & ,
PolyGamma[0, -#1]/(2*#1^2 +
4*#1 + 7) & ]
s // N // Chop
0.206647
s // ToRadicals // Simplify
(1/6)*(-PolyGamma[0, 1 - I] -
PolyGamma[0, 1 + I] +
PolyGamma[0, 1 - 2*I] +
PolyGamma[0, 1 + 2*I])
% // N // Chop
0.206647
s // ToRadicals // FullSimplify
(1/6)*(-HarmonicNumber[-I] -
HarmonicNumber[I] +
HarmonicNumber[-2*I] +
HarmonicNumber[2*I])
% // N // Chop
0.206647
NSum[k/((k^2 + 1) (k^2 + 4)), {k, 1, Infinity}]
0.206647
Bob Hanlon
---- "Chris H. Fleming" <chris_h_fleming at yahoo.com> wrote:
=============
Sum[k/((k^2 + 1) (k^2 + 4)), {k, 1, \[Infinity]}]
Sum does not converge.
NSum[k/((k^2 + 1) (k^2 + 4)), {k, 1, \[Infinity]}]
0.206647
Fortunately I know how to do this sum by hand, but Mathematica can
usually handle these Harmonic number functions pretty well.
Does anyone know a way of massaging this into a form Mathematica can
digest?