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We are asked to test the series for convergence using the **Ratio Test**. The given series is:

\[
\sum_{n=1}^{\infty} \frac{n^5}{0.5^n}
\]

The Ratio Test involves computing the limit:

\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]

where \(a_n = \frac{n^5}{0.5^n}\). Let's break it down step-by-step.

1. **Define \(f(n)\)**:
   Here, \(a_n = \frac{n^5}{0.5^n} = \frac{n^5}{\left(\frac{1}{2}\right)^n} = n^5 \cdot 2^n\).

So \(f(n) = \frac{a_{n+1}}{a_n} = \frac{(n+1)^5 \cdot 2^{n+1}}{n^5 \cdot 2^n}\).

2. **Simplify \(f(n)\)**:
   \[
   f(n) = \frac{(n+1)^5 \cdot 2^{n+1}}{n^5 \cdot 2^n} = \frac{(n+1)^5}{n^5} \cdot 2 = \left(\frac{n+1}{n}\right)^5 \cdot 2 = \left(1 + \frac{1}{n}\right)^5 \cdot 2
   \]

3. **Take the limit as \(n \to \infty\)**:
   \[
   \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^5 = 1
   \]
   Thus, the limit becomes:
   \[
   \lim_{n \to \infty} f(n) = 1 \cdot 2 = 2
   \]

4. **Apply the Ratio Test**:
   The Ratio Test states that if the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit equals 1, the test is inconclusive.

   Since the limit is 2, which is greater than 1, the series **diverges**.

---

Would you like more details on any step or have further questions?

Here are 5 related questions:
1. How does the Ratio Test work for other types of series?
2. What happens if the limit in the Ratio Test equals 1?
3. Can the Ratio Test be used for alternating series?
4. How does the series \(\sum n^k \cdot r^n\) generally behave for different values of \(k\) and \(r\)?
5. What is the difference between the Ratio Test and the Root Test?

**Tip**: Always simplify the ratio of consecutive terms before taking limits to avoid errors in convergence tests.

Test the series below for convergence using the Ratio Test: sum(n=1 to infinity) n^5 / (0.5^n). The limit of the ratio test simplifies to lim (n -> infinity) |f(n)| where f(n) is to be determined.

Explore how to test for series convergence using the Ratio Test with a series of the form n^5 / (0.5^n). This step-by-step solution applies key concepts of series convergence, limits, and the Ratio Test, providing a detailed breakdown suitable for undergraduate-level calculus students.

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