# Homework Help: Inverting an Asymptotic Equality

1. Oct 24, 2011

### jgens

1. The problem statement, all variables and given/known data

Show that the following two statements are equivalent:
1. $a_n^2\log{(a_n)} \sim n$
2. $a_n \sim \sqrt{\frac{2n}{\log{(n)}}}$

2. Relevant equations

N/A

3. The attempt at a solution

I can show $(2) \implies (1)$ pretty easily. However, I am having some difficulties with the proof in the other direction. I can get the proof down to the point where $a_n^2\log{(a_n)} = b_n^2\log{(b_n)}$ where $b_n = \sqrt{\frac{2n}{\log{(n)}}}$ but I don't know how to make the final step concluding $a_n \sim b_n$. Could anyone give me a little help with this?

Thanks