Testing for convergence in Infinite series with factorial in numerator I have the following infinite series that I need to test for convergence/divergence:
$$\sum_{n=1}^{\infty} \frac{n!}{1 \times 3 \times 5 \times \cdots \times (2n-1)}$$
I can see that the denominator will eventually blow up and surpass the numerator, and so it would seem that the series would converge, but I am not sure how to test this algebraically given the factorial in the numerator and the sequence in denominator. The recursive function for factorial $n! = n \times (n-1)!$ doesn't seem to simplify things in this case, as I cannot eliminate the $(2n-1)$ in the denominator. Is there a way to find a general equation for the denominator such that I could perform convergence tests (e.g. by taking the integral, limit comparison, etc.)

Note that
$$
\frac{n!}{(2n-1)!!}=\frac11\frac23\frac35\frac47\cdots\frac{n}{2n-1}
$$
For $n\ge2$, $\dfrac{n}{2n-1}\le\dfrac23$. Thus, for $n\ge1$, we have
$$
\frac{n!}{(2n-1)!!}\le\left(\frac23\right)^{n-1}
$$
We can then compare with a geometric series.
