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Re: real telescopic sum becomes complex?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg54408] Re: [mg54358] real telescopic sum becomes complex?
*From*: Bob Hanlon <hanlonr at cox.net>
*Date*: Sun, 20 Feb 2005 00:08:13 -0500 (EST)
*Reply-to*: hanlonr at cox.net
*Sender*: owner-wri-mathgroup at wolfram.com
$Version
5.1 for Mac OS X (October 25, 2004)
telesum[n_]:=Sum[Sqrt[k]-Sqrt[k-1],{k,1,n}];
Table[telesum[n],{n,10}]
{1, Sqrt[2], Sqrt[3], 2, Sqrt[5], Sqrt[6], Sqrt[7],
2*Sqrt[2], 3, Sqrt[10]}
telesum[n]
HarmonicNumber[n, -(1/2)] - HarmonicNumber[n - 1, -(1/2)]
%/.n->Range[10]
{1, Sqrt[2], Sqrt[3], 2, Sqrt[5], Sqrt[6], Sqrt[7],
2*Sqrt[2], 3, Sqrt[10]}
Bob Hanlon
>
> From: Peter Pein <petsie at arcor.de>
To: mathgroup at smc.vnet.net
> Date: 2005/02/19 Sat AM 02:32:02 EST
> To: mathgroup at smc.vnet.net
> Subject: [mg54408] [mg54358] real telescopic sum becomes complex?
>
> Dear Group,
>
> I'v only got the old version 4.0 of Mathematica and I would like to know
> whether later versions do the same mistake:
>
> In[1]:=
> {$Version, $ProcessorType}
> Out[1]=
> {"4.0 for Microsoft Windows (July 16, 1999)", "x86"}
> In[2]:=
> telesum[n_] := Sum[Sqrt[k] - Sqrt[k - 1], {k, 1, n}]
> In[3]:=
> telesum[5]
> Out[3]=
> Sqrt[5]
> OK - but:
> In[4]:=
> telesum[n]
> Out[4]=
> HarmonicNumber[n, -(1/2)] +
> I*(Zeta[-(1/2)] - Zeta[-(1/2), 1 - n])
> In[5]:=
> % /. n -> 5
> Out[5]=
> 3 + Sqrt[2] + Sqrt[3] + Sqrt[5] +
> I*(Zeta[-(1/2)] - Zeta[-(1/2), -4])
> In[6]:=
> N[%]
> Out[6]=
> 8.382332347441762 - 6.146264369941973*I
>
> (please note, that Re[%]+Im[%]==Sqrt[5.]!!!)
>
> Has anyone got an explanation, what could have happened inside
> Mathematica to get this nonsense?
>
> Thanks in advance,
> Peter
>
> --
> Peter Pein
> Berlin
>
>
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