The problem is asking to find the radius of convergence for the series

$\sum_{n=1}^\infty \frac{2 \cdot 4 \cdot 6 \cdots (2n)}{(2n)!}x^n.$

Step 1: Simplify the numerator

The numerator $2 \cdot 4 \cdot 6 \cdots (2n)$ is the product of all even numbers from 2 to $2n$. This can be written as:

$2^n \cdot n!,$

because each term in the product is $2 \cdot 1, 2 \cdot 2, 2 \cdot 3, \dots, 2 \cdot n$.

So the series becomes:

$\sum_{n=1}^\infty \frac{2^n \cdot n!}{(2n)!}x^n.$

Step 2: Ratio Test

To find the radius of convergence, we use the Ratio Test. Let $a_n = \frac{2^n \cdot n!}{(2n)!}$. The Ratio Test examines the limit:

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

Substitute $a_n$ into the ratio:

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

Step 3: Simplify the ratio

1. $2^{n+1} / 2^n = 2$,
2. $(n+1)! / n! = (n+1)$,
3. The factorial ratio $(2n)! / (2(n+2))! = \frac{1}{(2n+2)(2n+1)}$.

So the ratio becomes:

$\frac{a_{n+1}}{a_n} = 2 \cdot (n+1) \cdot \frac{1}{(2n+2)(2n+1)}.$

Simplify further:

$\frac{a_{n+1}}{a_n} = \frac{2(n+1)}{(2n+2)(2n+1)} = \frac{2}{(2n+1)}.$

Step 4: Take the limit

As $n \to \infty$:

$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{2}{(2n+1)} = 0.$

This limit is less than 1 for all $x$, so the series converges for all $x$. Hence, the radius of convergence is:

$\boxed{\infty}.$

Would you like further clarification or a breakdown of specific steps?

Here are 5 related questions to explore further:

1. What is the convergence interval for a series with infinite radius of convergence?
2. How does the factorial in the denominator affect the convergence of a series?
3. Can you generalize this series for odd factorial terms?
4. What role does the Ratio Test play in finding the radius of convergence?
5. What are some other methods to determine the radius of convergence besides the Ratio Test?

Tip: When working with factorials and products, try to rewrite them in terms of simpler expressions, like powers or factorial ratios, to simplify calculations.