To find the interval of convergence of the series

$\sum_{n=1}^{\infty} \frac{(x-4)^n}{(2n)!},$

we can use the Ratio Test. The Ratio Test states that for a series $\sum a_n$, the series converges if

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

Let's apply this to our series. Here,

$a_n = \frac{(x-4)^n}{(2n)!}.$

We need to find the limit of

$\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{(x-4)^{n+1}}{(2(n+1))!}}{\frac{(x-4)^n}{(2n)!}} \right|.$

Simplifying the fraction inside the absolute value, we get:

$\left| \frac{(x-4)^{n+1}}{(2(n+1))!} \cdot \frac{(2n)!}{(x-4)^n} \right| = \left| \frac{(x-4)^{n+1}}{(x-4)^n} \cdot \frac{(2n)!}{(2n+2)!} \right| = \left| (x-4) \cdot \frac{1}{(2n+1)(2n+2)} \right|.$

As $n$ approaches infinity, the term $\frac{1}{(2n+1)(2n+2)}$ approaches zero. Therefore,

$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| (x-4) \cdot \frac{1}{(2n+1)(2n+2)} \right| = 0.$

Since the limit is zero for all $x$, which is less than 1, the series converges for all $x$.

Therefore, the interval of convergence is:

$\boxed{-\infty < x < \infty}.$

Would you like more details or have any questions?

Here are some additional questions to deepen your understanding:

1. How does the Ratio Test determine the interval of convergence?
2. What happens if the limit in the Ratio Test is greater than 1?
3. Can you provide an example of a series with a finite interval of convergence?
4. How do you apply the Ratio Test to a series involving factorial terms?
5. What is the difference between absolute convergence and conditional convergence?
6. How does the interval of convergence affect the behavior of a power series?
7. Can you use other tests to determine the interval of convergence for this series?
8. What are some applications of power series in mathematical analysis?

Tip: When applying the Ratio Test, carefully handle factorials and powers, and ensure you correctly simplify the limit expression to accurately determine the interval of convergence.