Recurrence relations in asymptotic regime

[dingo_d]

Homework Statement

I'm solving the quantum harmonic oscillator. And I'm solving Schrödinger equation. So I came up to one part where I have to use power series method of solving DE (that or Frobenius would probably work just fine). Now I have the recurrence relation:

$$a_{n+2}=\frac{\lambda(2n+1)-k^2}{(n+2)(n+1)}a_n$$

And the text in which this is solved says that for $$n\ton\infty$$ that leads to asymptotic law

$$a_{n+2}=\frac{2\lambda}{n}a_n$$ corresponding to the series expansion of $$e^{\lambda x^2}$$.

Now, I tried looking at the limit, via L'Hospitals rule and I really can't see how they got that! :\

So can someone explain to me how they got that? Thanks...

 

[hunt_mat]

Expand and divide top and bottom by n:
$$a_{n+1}=\frac{\lambda\left( 2+\frac{1}{n}\right)-\frac{k^{2}}{n}}{n+3+\frac{1}{n}}a_{n}$$
For n large, the terms in 1/n can be ignored

 

[dingo_d]

And I just ignore the 3 in the denominator can be ignored because when n goes to infinity that is negligible, right?

 

[hunt_mat]

Yep, sound good to me.

 

[dingo_d]

Thanks ^^