       Re: Problem with a sum

• To: mathgroup at smc.vnet.net
• Subject: [mg53923] Re: Problem with a sum
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Fri, 4 Feb 2005 04:11:19 -0500 (EST)
• Organization: The University of Western Australia
• References: <ctniaq\$evd\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <ctniaq\$evd\$1 at smc.vnet.net>, ncc1701zzz at hotmail.com wrote:

> I would like to ask you a question about a sum in a problem I have
> found in Mathematica 5.1.
>
> The sum is the following:
>
> Sum[(k^2 - (1/2))/(k^4 + (1/4)), {k, 1, 1000}]
>
> I have no problems with the sum in that form, but the following one
> doesn't work:
>
>
> s=Sum[(k^2 - (1/2))/(k^4 + (1/4)), {k, 1, m}]
> s /. m->1000
>
>
> It gives a long result with hypergeometric functions. Also, it cannot
> be converted to a number with N[], due to some kind of ComplexInfinity
> problem. FullSimplify doesn't help, neither.

Here is one approach: first generalize the problem, putting a for 1/2
and b for 1/4.

s[m_][a_, b_] = Sum[(k^2 - a)/(k^4 + b), {k, 1, m}];

Here is the numerical value of the sum to 1000 terms

N[s[1/2, 1/4], 40]

> Also, if I evaluate it to Infinity, I cannot get the value symbolically
> nor numerically, except if I use NSum[], that gives me the right
> result, 1.

The sum to infinity is somewhat simpler:

s[Infinity][a_, b_] = Sum[(k^2 - a)/(k^4 + b), {k, 1, Infinity}]

N[s[Infinity][1/2, 1/4], 40]

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 6488 2734
School of Physics, M013                         Fax: +61 8 6488 1014
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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