Thread: Limit (~ is Landau's symbol in this Qs)

1. Limit (~ is Landau's symbol in this Qs)

Hi everyone:
I got a question which is about limit, after thinking still don't know how to starts it.

Qrove that (n+a)!/(n+b)! ~ n^(a-b) as n goes to +ive infinite (for fixed a,b>0)

and the Qs provide some formula to use
Euler's limit: '[1+(x/n)]^n' → 'e^x' as n goes to infinite
Stirling's formula: n! ~ [root(2πn)](n/e)^n

Thanks very much.

2. Originally Posted by wsun
Hi everyone:
I got a question which is about limit, after thinking still don't know how to starts it.

Qrove that (n+a)!/(n+b)! ~ n^(a-b) as n goes to +ive infinite (for fixed a,b>0)

and the Qs provide some formula to use
Euler's limit: '[1+(x/n)]^n' → 'e^x' as n goes to infinite
Stirling's formula: n! ~ [root(2πn)](n/e)^n

Thanks very much.
Start by using Stirling's formula to replace the factorials in $\displaystyle (n+a)!/(n+b)!$

(you might want to consider why what you know means:

$\displaystyle \frac{(n+a)!}{(n+b)!} \sim \frac{\sqrt{2\pi (n+a)}((n+a)/e)^{n+a}}{\sqrt{2\pi (n+b)}((n+b)/e)^{n+b}}$ $\displaystyle \sim \frac{((n+a)/e)^{n+a}}{((n+b)/e)^{n+b}}$

Then use the other limit)

CB

3. I got it.
Thanks