Re: Mathematical Question

[Ada Cheng]

You can write (1-x^c)=(1-x)(1+x+...+x^{c-1}), this allows you to rewrite
y as (sum_{k=1}^c x^k-x^c)/(1-x^c). y is clearly now in indeterminate
form as x approaches 1.  Apply L'Hopital rule, do some algebra and you
should get the answer.  (to make life easy on yourself, you will want to
use finite sum of arithmetic series formula).

Hope this help.

Ada
On Mon, 2 Aug 2004, Henry F. Mollet wrote:

> I have nowhere else to turn, I'm hoping the octave help
> list can provide a tip. Many thanks, Henry
>
> y = x/(1-x) - cx^c/(1-x^c),
> where c is a positive integer constant
>
> I need to know y in the limit as x approaches 1.
> I even have the result by biological reasoning
> and it is (c-1)/2 but how do I prove it mathematically?
>
> For example using x = 0.999 and c = 100 we have:
> y = 999 - 950.4 = 48.6 whereas (c-1)/2 = 49.5;
> Using x = 0.9999 and same c = 100, we have:
> y = 9999 - 9949.6 = 49.4 whereas (c-1)/2 = 49.5, close enough.
>
> Somehow I have to expand the second term in the
> expression into a series where the first term of the
> series will be the same as the first term of the
> expression (i.e. x/(1-x)) and they will cancel out.
>
> In my application, the constant c is a positive integer,
> and that's what I used for empirical checks of the result
> but I have not checked if that's a requirement.
>
>
>
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============================================================================
 Ada Cheng                              address@hidden
 Assistant Professor                    http://www.kettering.edu/~acheng
 Department of Science and Mathematics
 Kettering University
 1700 West Third Avenue
 Flint, Michigan 48504-4898
 U.S.A.
============================================================================



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