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)
• 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
>
>
> Has anyone got an explanation, what could have happened inside
> Mathematica to get this nonsense?
>
>   Peter
>
> --
> Peter Pein
> Berlin
>
>

```

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