Math Is Fun Forum

nha

Member

Registered: 2010-09-05

Posts: 43

Catalan number help

Hey everyone, I need some more help dealing with Catalan number function.

where

denotes the

-Catalan number.

I know I need to use Stirling's estimate:

for integers

and the limit definition of

, but I just can't seem to make anything of it.

Thanks to anyone that helps.

Last edited by nha (2010-09-05 18:30:00)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Catalan number help

Hi nha;

I am not following you here. To get the n th Catalan Number you just plug into that poor approximation.

There is an exact formula for Catalan numbers and a generating function.

Are you trying to prove  that the ratio of that approximation to the n th Catalan number approaches 1 as n->∞?
That is what you need stirlings approximation for.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

nha

Member

Registered: 2010-09-05

Posts: 43

Re: Catalan number help

Hi nha;

I am not following you here. To get the n th Catalan Number you just plug into that poor approximation.

There is an exact formula for Catalan numbers and a generating function.

Are you trying to prove  that the ratio of that approximation to the n th Catalan number approaches 1 as n->∞?
That is what you need stirlings approximation for.

lol, I must have deleted "Prove that for m very large".

The question is:
Prove that for m very large

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Catalan number help

Hi nha;

One way is;

This is the formula for the nth catalan number.

Form the ratio between it and your formula and take the limit.

Now using stirlings for the denominator.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

nha

Member

Registered: 2010-09-05

Posts: 43

Re: Catalan number help

Ah yes, thanks again bobbym. I see why I couldn't get it right.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Catalan number help

Hi nha;

I cleaned it up a little bit. Glad to help!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.