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Re: Expressions with ellipsis (...)
*To*: mathgroup at smc.vnet.net
*Subject*: [mg93015] Re: Expressions with ellipsis (...)
*From*: "David Park" <djmpark at comcast.net>
*Date*: Fri, 24 Oct 2008 02:27:22 -0400 (EDT)
*References*: <gdmsbd$jv$1@smc.vnet.net>
If I understand your question correctly you are solving the following
problem:
HoldForm[Sum[(n - i)^2, {i, 0, n - 1}]/n^3]
% // ReleaseHold
Limit[%, n -> Infinity]
\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(n - 1\)]
\*SuperscriptBox[\((n - i)\), \(2\)]\)/n^3
((1 + n) (1 + 2 n))/(6 n^2)
1/3
But you would prefer that the initial expression was an ellipsis expression
in the form of the standard textbook representation. This can be done using
the functionality of the Idenfinite Sequences section of the Presentation
package. This is how it would be done:
Needs["Presentations`Master`"]
iSum[i^2, {i, n, HoldForm[n - 1], HoldForm[n - 2], , 1}]/n^3
% // iSumNormalize[tag1, {i, n, 1}]
Limit[%, n -> Infinity]
(n^2 + (n-1)^2 + (n-2)^2 + ... + 1)/n^3 (but in built up box form)
((1 + n) (1 + 2 n))/(6 n^2)
1/3
iSum represents an ellipsis expression where the sequence of terms represent
the explicit terms you wish to display and you can have any number of
ellipses, which are entered as Nulls in the sequence. iSumNormalize will
convert the iSum to a regular Sum for evaluation by Mathematica. It just
substitutes a normal iterator. It is also possible to operate on the terms
while they are still in the ellipsis expression. There are also ellipsis
expressions for Products, Tables, Sequences and expressions with arbitrary
Infixes.
One of the main purposes of the Presentations package is to make it easier
to use Mathematica in a 'textbook mode' and to work with standard textbooks.
It has over 5 MB of code and help pages, but is probably far beyond the
budget of most students.
--
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/
"dch888" <dch888 at googlemail.com> wrote in message
news:gdmsbd$jv$1 at smc.vnet.net...
> 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.
>
>
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