This is an infinite series:
$\sum_{n=0}^\infty n! (x + 3)^n$

Step-by-Step Solution:

1. Analyze the series terms: Each term in the series is of the form $n! (x + 3)^n$.

2. Behavior when $n = 0$: When $n = 0$: $n! (x + 3)^n = 0! (x + 3)^0 = 1 \cdot 1 = 1$

   So the first term is $1$.

3. Convergence Test: Since the factorial $n!$ grows very fast for $n > 1$, this series diverges for all $x \neq -3$, because the term $n! (x + 3)^n$ grows without bound.

4. Conclusion: The series diverges for all $x \neq -3$. For $x = -3$, the series converges trivially to $1$, as all terms except the first become $0$.

Would you like a detailed explanation of the convergence test, or any other details?

Here are 5 related questions for further exploration:

1. What is the general form of a factorial, and how does it grow as $n \to \infty$?
2. How can the Ratio Test be applied to determine the convergence of series?
3. What happens to the series if we replace $n!$ with $\frac{1}{n!}$?
4. What is the domain of convergence for power series in general?
5. Can the series $\sum_{n=0}^\infty a_n (x + c)^n$ converge for some $a_n$? How does $a_n$ affect convergence?

Tip: When working with infinite series, testing convergence is key to understanding its behavior.