https://file.aiquickdraw.com/mathbot/file/1731914657601p3bmlk5l.jpg

To analyze this problem and provide a solution, I'll work through the **Ratio Test** for each series. The Ratio Test states that for a series \( \sum a_n \):

- If \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \):
  - \( L < 1 \): the series converges.
  - \( L > 1 \) or \( L = \infty \): the series diverges.
  - \( L = 1 \): the test is inconclusive.

Let me process and determine for which of the given series the ratio test ensures **divergence**.from sympy import symbols, factorial, exp, oo, simplify, limit

# Define the variable n
n = symbols('n', integer=True, positive=True)

# Define each term of the series
series_terms = {
    "Series 1": factorial(n) / (2**n + 3),
    "Series 2": ((n + 1) * 2**n) / factorial(n + 1),
    "Series 3": factorial(n) / factorial(2*n),
    "Series 4": (factorial(n)**2) / factorial(2*n),
    "Series 5": exp(n) / (n**2)
}

# Compute ratio test for each series
results = {}
for name, term in series_terms.items():
    ratio = simplify((term.subs(n, n+1)) / term)  # a_{n+1} / a_n
    limit_ratio = limit(ratio, n, oo)  # Take the limit as n approaches infinity
    results[name] = limit_ratio

results{'Series 1': oo, 'Series 2': 0, 'Series 3': 0, 'Series 4': 1/4, 'Series 5': E}{'Series 1': oo, 'Series 2': 0, 'Series 3': 0, 'Series 4': 1/4, 'Series 5': E}successHere are the results of applying the Ratio Test to the series:

1. **Series 1** (\( \frac{n!}{2^n + 3} \)): The ratio test yields \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \infty \), indicating **divergence**.

2. **Series 2** (\( \frac{(n + 1) \cdot 2^n}{(n + 1)!} \)): The ratio test yields \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 0 \), indicating **convergence**.

3. **Series 3** (\( \frac{n!}{(2n)!} \)): The ratio test yields \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 0 \), indicating **convergence**.

4. **Series 4** (\( \frac{(n!)^2}{(2n)!} \)): The ratio test yields \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \frac{1}{4} \), indicating **convergence**.

5. **Series 5** (\( \frac{e^n}{n^2} \)): The ratio test yields \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = e \), indicating **divergence**.

### Conclusion
The Ratio Test ensures **divergence** for:
- **Series 1**
- **Series 5**

Let me know if you want further clarifications or explanations!

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### Expanding Questions:
1. How does the ratio test differ from other convergence tests like the root test?
2. Can you explain why the factorial growth in Series 1 leads to divergence?
3. For Series 4, why does a limit of \( \frac{1}{4} \) confirm convergence?
4. How would you approach the divergence of Series 5 with another convergence test?
5. What are the implications of \( L = 1 \) in the ratio test?

### Tip:
For factorial-based series, factorial growth often dominates exponential or polynomial terms, so be mindful of their rapid divergence.

For which of the following series does the ratio test ensure divergence? Select all that apply.

Learn how to determine divergence of series using the Ratio Test. This problem involves factorials, exponential terms, and powers to test convergence. Includes step-by-step application of the ratio test and interpretations of results.

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