[Math] Testing for convergence in Infinite series with factorial in numerator

Question

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.)

Answer 1

Hint: Use the Ratio Test. No modification of the expression is needed.