Does the sequence $a_n = \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)}$ converge?
This is a homework question for an analysis class.
Firstly, I know the sequence is comprised of odd numbers up to $2n-1$ over even numbers up to $2n$.
Secondly, I know the sequence decreases. I'm guessing it'll converge, but not specifically to $0$.
Where I am stuck is simplifying the series to something easier to work with. Any help would be greatly appreciated!
Edit: solved, thank you everyone. I ended up doing what was first suggested, not finding the actual limit but just proving the sequence converges. Thanks again!