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To determine the convergence or divergence of the series 

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

using the Ratio Test, let's analyze the setup:

1. First, identify the general term \( a_n = \frac{1}{5^n} \).

2. The Ratio Test states that for a series with terms \( a_n \), we calculate the limit

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

   If \( L < 1 \), the series converges absolutely. If \( L > 1 \) or \( L = \infty \), the series diverges. If \( L = 1 \), the test is inconclusive.

### Step 1: Set up the ratio \( \frac{a_{n+1}}{a_n} \)

For our series:
\[
a_n = \frac{1}{5^n}
\]
and
\[
a_{n+1} = \frac{1}{5^{n+1}}
\]

### Step 2: Calculate \( \frac{a_{n+1}}{a_n} \)

\[
\frac{a_{n+1}}{a_n} = \frac{\frac{1}{5^{n+1}}}{\frac{1}{5^n}} = \frac{1}{5}
\]

### Step 3: Take the limit

Since \( \frac{a_{n+1}}{a_n} = \frac{1}{5} \) for all \( n \), we get

\[
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{5}
\]

### Conclusion

Since \( L = \frac{1}{5} < 1 \), the Ratio Test tells us that the series converges.

---

Would you like more details or have questions about any steps?

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Here are some related questions to deepen understanding:

1. How does the Ratio Test compare to other convergence tests, like the Root Test?
2. What would happen if the base of the exponent were a different number, like 10, instead of 5?
3. Can we conclude absolute convergence for all values of \( L < 1 \), or are there exceptions?
4. How can we apply the Ratio Test to series with factorial terms, such as \( \sum \frac{n!}{n^n} \)?
5. Why is the Ratio Test inconclusive when \( L = 1 \)?

**Tip:** Always remember that the Ratio Test is particularly useful for series involving exponential and factorial terms, where ratios simplify the analysis effectively.

Use the Ratio Test to determine the convergence or divergence of the series ∑_{n=1}^{∞} (1 / 5^n).

Learn how to use the Ratio Test to determine if the series ∑_{n=1}^{∞} (1 / 5^n) converges or diverges. This detailed solution includes steps to apply the Ratio Test, which is essential for analyzing series with exponential terms. Ideal for high school students in Grades 11-12.

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