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Re: Limit[(p*(-1 + p^n))/(-1 + p), n->Infinity]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg28928] Re: [mg28918] Limit[(p*(-1 + p^n))/(-1 + p), n->Infinity]
*From*: andrzej <andrzej at bekkoame.ne.jp>
*Date*: Sat, 19 May 2001 22:27:48 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
There is one trick, which often works in such cases and which I have
described several times in this group in the past. The idea is to
replace the limit of a sequence a[n] by the infinite sum
Sum[a[n+1]-a[n],{n,1,Infinity}]+a[1]. Mathematica can quite frequently
cope with such expressions symbolically. This is one of these cases:
In[2]:=
Simplify[Sum[(p*(p^(n + 1) - 1))/(p - 1) -
(p*(p^n - 1))/(p - 1), {n, 1, Infinity}] + p]
Out[2]=
p
-----
1 - p
Note that Mathematica here makes a silent assumption that -1<p<1. It
sees to me somewhat questionable if such silent assumptions are a good
thing (I would prefer answers in the form If[....]) but this happens
quite often in Mathematica and seems to me to be a legacy from early
versions (before Assumptions and Domains).
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
http://sigma.tuins.ac.jp/~andrzej/
On Friday, May 18, 2001, at 02:13 PM, Adalbert Hanssen wrote:
> Hi, MathGroup,
>
> of course, this one is easily done by hand:
>
> Limit[(p*(-1 + p^n))/(-1 + p), n->Infinity]
>
> but how does one do it with Mathematica?
> Of course, 0<=p<=1 (real) and n runs through integers.
>
>
> kind regards
>
> Dipl.-Math. Adalbert Hanszen
>
>
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