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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>
  • Reply-to: murray at math.umass.edu

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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