The problem asks to determine whether the given series converges or diverges using the ratio test (kvotientkriteriet in Danish). The series provided is:

$\sum_{n=1}^{\infty} (-1)^n \cdot \frac{(n+2)! \cdot n^2}{n! \cdot 3^{2n}}$

Applying the Ratio Test:

The ratio test for convergence involves computing the limit of the absolute value of the ratio of successive terms. Let $a_n = (-1)^n \cdot \frac{(n+2)! \cdot n^2}{n! \cdot 3^{2n}}$, and we will calculate the following:

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

First, let's express $a_{n+1}$:

$a_{n+1} = (-1)^{n+1} \cdot \frac{(n+3)! \cdot (n+1)^2}{(n+1)! \cdot 3^{2(n+1)}}$

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

$\frac{a_{n+1}}{a_n} = \frac{(-1)^{n+1} \cdot \frac{(n+3)! \cdot (n+1)^2}{(n+1)! \cdot 3^{2(n+1)}}}{(-1)^n \cdot \frac{(n+2)! \cdot n^2}{n! \cdot 3^{2n}}}$

Simplify the expression:

1. Factorial terms: $\frac{(n+3)!}{(n+2)!} = (n+3)$ and $\frac{(n+1)!}{n!} = (n+1)$

2. Powers of 3: $\frac{3^{2n}}{3^{2(n+1)}} = \frac{1}{9}$

3. Remaining terms: $\frac{(n+1)^2}{n^2}$

Now substitute these back into the ratio:

$\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(n+3)(n+1)^2}{(n+1) \cdot n^2} \cdot \frac{1}{9} \right|$

Simplifying further:

$\left| \frac{a_{n+1}}{a_n} \right| = \frac{(n+3)(n+1)}{n^2} \cdot \frac{1}{9}$

As $n \to \infty$, this expression approaches:

$L = \lim_{n \to \infty} \frac{(n+3)(n+1)}{n^2} \cdot \frac{1}{9} = \frac{1}{9}$

Conclusion:

Since $L = \frac{1}{9} < 1$, the ratio test confirms that the series converges.

Would you like a more detailed explanation of any part of the process?

Here are 5 follow-up questions:

1. What happens if the limit in the ratio test is greater than 1?
2. How does the alternating sign affect the convergence of the series?
3. Can we apply other tests to determine convergence for this series?
4. How does the factorial growth compare to the exponential terms in this problem?
5. What is the significance of $(-1)^n$ in alternating series?

Tip: The ratio test is particularly useful for series involving factorials or exponentials, as it simplifies the terms significantly.