       Re: Expressions with ellipsis (...)

• To: mathgroup at smc.vnet.net
• Subject: [mg93017] Re: [mg92995] Expressions with ellipsis (...)
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Fri, 24 Oct 2008 02:27:45 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <200810220939.FAA00651@smc.vnet.net>

```If you just want to evaluate this, then it's easy since, among other
things, there's a well-know closed-form expression for the sum of
squares of the first n positive integers:

Sum[(n - i)^2, {i, 0, n - 1}]
(n*(1 + n)*(1 + 2*n))/6

(I'm showing InputForm for the output here rather than the usual
OutputForm that would show fractions build up in 2 dimensions.)

And so:

Limit[%/n^3,n->Infinity]
1/3

Or, of course, the whole thing together:

Limit[Sum[(n - i)^2, {i, 0, n - 1}]/n^3, n->Infinity]
1/3

If you want to DISPLAY such an "indefinite sum", then David Park's
"Presentations" application package helps:

Needs["Presentations`Master`"]
iSum[(n - i)^2, {i, 0, 1, 2, , n - 1}]
n^2 + (-1+n)^2 + (-2+n)^2 + ... + 1

(literally just like that, except that the exponents are shown as
superscripts).

dch888 wrote:
> Hi Mathematica Friends,
>
> I want to do this:
>
> Limit[(n^2 + (n - 1)^2 + (n - 2)^2 + ... + 1)/n^3, n -> Infinity]
>
> But Mathematica barfs:
>
>      expression cannot be followed by "...".
>
> Searching the help for 'ellipsis' gives me the Unicode 2026 character
> (i.e. ...).
>
> How can I get Mathematica to eval this limit?
>
> Thanks,
> David.
>
>

--
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

```

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