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zhangyi.zhangyi@gmail.com
science forum beginner

Joined: 10 Jul 2006
Posts: 1

Posted: Mon Jul 10, 2006 11:32 am    Post subject: a limit problem

Define that (n)_r=n(n-1)...(n-r+1),
what's the result of the limit:
lim_{n->\infty, r->\infty, r/n->p} (n-k)_r / (n)_r.

is it e^{-kp} ?
how to prove it?
Einar Andreas Rødland
science forum beginner

Joined: 03 Jun 2005
Posts: 10

Posted: Mon Jul 10, 2006 4:09 pm    Post subject: Re: a limit problem

zhangyi.zhangyi@gmail.com wrote:
 Quote: Define that (n)_r=n(n-1)...(n-r+1), what's the result of the limit: lim_{n->\infty, r->\infty, r/n->p} (n-k)_r / (n)_r. is it e^{-kp} ? how to prove it?

First, using (n)_r = n!/(n-r)!, you see that
(n-k)_r / (n)_r = (n-r)_k / (n)_k.
Write this out, and you get
(n-r)/n * (n-r-1)/(n-1) * ... * (n-r-k+1)/(n-k+1).
The limit as n,r increases with r/n -> p is then
(1-p) * (1-p) * ... * (1-p) = (1-p)^k.

Einar

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