To solve this problem correctly using the Ratio Test, let's proceed through each part in detail.

1. Identify $a_n$: Based on the image, you've already identified $a_n$ as: $a_n = \frac{(-1)^{n-1} n!}{1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n-1)}$ This part is correct.

2. Set up the Ratio Test: The Ratio Test states that for a series $\sum a_n$, if we compute $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L,$ then:

   • If $L < 1$, the series is convergent.
   • If $L > 1$, the series is divergent.
   • If $L = 1$, the test is inconclusive.

3. Evaluate $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$: Let’s compute this limit carefully.

   Start by finding $a_{n+1}$: $a_{n+1} = \frac{(-1)^n (n+1)!}{1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n+1)}.$

   Now, calculate the ratio $\frac{a_{n+1}}{a_n}$: $\frac{a_{n+1}}{a_n} = \frac{\frac{(-1)^n (n+1)!}{1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n+1)}}{\frac{(-1)^{n-1} n!}{1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n-1)}}.$

   Simplifying this expression: $\frac{a_{n+1}}{a_n} = \left| \frac{(-1) \cdot (n+1)!}{n! \cdot (2n+1)} \right| = \frac{n+1}{2n+1}.$

4. Take the Limit: Now, evaluate the limit as $n \to \infty$: $\lim_{n \to \infty} \frac{n+1}{2n+1} = \lim_{n \to \infty} \frac{n(1 + \frac{1}{n})}{2n(1 + \frac{1}{2n})} = \frac{1}{2}.$

5. Conclusion Using the Ratio Test: Since $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{2} < 1$, the series is convergent.

Final Answer

• Since the limit is less than 1, the series converges by the Ratio Test.

Would you like further details or have any questions about these steps?

Here are five related questions that expand on this topic:

1. How does the Ratio Test compare to the Root Test for determining convergence?
2. What other convergence tests could be applied to this series, and why?
3. How would the Ratio Test be affected if the terms of the series had a different factorial structure?
4. What are some common mistakes to avoid when applying the Ratio Test to factorial-based series?
5. Can the Ratio Test be applied to a series with terms involving powers instead of factorials?

Tip: When working with factorials in convergence tests, simplify expressions carefully, especially when taking limits at infinity.