In my question about the convergence/divergence of
$$ \sum_{n=2}^\infty \frac{1\cdot 3\cdot 5\cdot 7\cdots (2n-3)}{2^nn!}. $$
here: Why Doesn't This Series Converge?
Zarrax gave the answer:
"You can use Taylor approximations here. Note that the ratio between consecutive terms is ${2n - 3 \over 2n} = \exp(\ln(1 - 3/2n)) = \exp(-{3 \over 2n} + O(1/n^2))$. So the product is comparable to $\exp(-{3 \over 2} \sum_{i = 2}^n {1 \over n} + O(1/n))$, which in turn is comparable to $\exp(-{3 \over 2} \ln(n))$ or $n^{-{3 \over 2}}$. Thus the series converges."
I've decided to give a talk in a graduate seminar about the danger of coming up with examples off the top of your head and now wish to understand what this answer means. The problem is that I have no exposure to big O notation and am not having luck online. Basically, I don't understand the answer at all. I can break it into a few questions:
How does Zarrax pass from $\exp(\ln(1 - 3/2n))$ to $\exp(-{3 \over 2n} + O(1/n^2))$?
To which product is Zarrax referring in his third sentence? And how is it comparable to $\exp(-{3 \over 2} \sum_{i = 2}^n {1 \over n} + O(1/n))$?
How does Zarrax pass from that to $\exp(-{3 \over 2} \ln(n))$?
Thank you for your help.