To find the radius of convergence of the series $\sum_{n=0}^{\infty} x^{2n} (2n)!$, we will again use the ratio test.

The general term of the series is:

$a_n = x^{2n} (2n)!$

Applying the ratio test:

$\frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

where:

$a_{n+1} = x^{2(n+1)} (2(n+1))! = x^{2n+2} (2n+2)!$

Now calculate the ratio $\frac{a_{n+1}}{a_n}$:

$\frac{a_{n+1}}{a_n} = \frac{x^{2n+2} (2n+2)!}{x^{2n} (2n)!}$

Simplify this expression:

$\frac{a_{n+1}}{a_n} = x^2 \cdot \frac{(2n+2)(2n+1)(2n)!}{(2n)!} = x^2 (2n+2)(2n+1)$

Now, find the limit as $n \to \infty$:

$\lim_{n \to \infty} \left| x^2 (2n+2)(2n+1) \right|$

Since $(2n+2)(2n+1) \to \infty$ as $n \to \infty$, we see that:

$\lim_{n \to \infty} \left| x^2 (2n+2)(2n+1) \right| = \infty \text{ if } x \neq 0$

Thus, the series diverges for any $x \neq 0$. The only value of $x$ for which this series converges is $x = 0$.

Therefore, the radius of convergence $R$ is:

$R = 0$

The series converges only when $x = 0$.

Would you like further details or have any questions? Here are some related topics you might find interesting:

1. What are the criteria for using the ratio test versus other convergence tests?
2. How does the factorial term affect the convergence of a series?
3. How to find the interval of convergence for a series?
4. What is the significance of the radius of convergence in power series?
5. How does Stirling's approximation help in analyzing factorial terms in series?

Tip: When working with factorial terms, consider their rapid growth rate, as they can dominate other terms and significantly influence convergence behavior.