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This problem involves using the **Root Test** to determine whether the given series converges or diverges. The series is:

\[
\sum_{n=1}^{\infty} \frac{1}{3^n}
\]

### The Root Test:
The Root Test involves calculating the following limit:

\[
L = \lim_{n \to \infty} \sqrt[n]{|a_n|}
\]

where \( a_n \) represents the terms of the series. The series will converge if \( L < 1 \), diverge if \( L > 1 \), and is inconclusive if \( L = 1 \).

### Step 1: Identifying \( a_n \)
In this series, \( a_n = \frac{1}{3^n} \).

### Step 2: Applying the Root Test
We need to compute:

\[
L = \lim_{n \to \infty} \sqrt[n]{\left| \frac{1}{3^n} \right|}
\]

This simplifies to:

\[
L = \lim_{n \to \infty} \frac{1}{3}
\]

Since the \(n\)-th root of \(3^n\) is just 3, and \( \frac{1}{3} \) is constant, the limit is simply:

\[
L = \frac{1}{3}
\]

### Step 3: Conclusion
Since \( L = \frac{1}{3} < 1 \), the series **converges** by the Root Test.

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Would you like further clarification on this solution or the steps involved?

Here are 5 related questions to expand your understanding:

1. How does the root test compare with the ratio test for convergence?
2. Can the root test be applied to all series? If not, when does it fail?
3. What is the significance of the \( L = 1 \) case in the Root Test?
4. What other methods could be used to determine the convergence of geometric series like this?
5. How does the \( n \)-th root simplify when dealing with powers of constants?

**Tip:** For geometric series, the common ratio \( r \) determines convergence. If \( |r| < 1 \), the series converges.

Use the Root Test to determine the convergence or divergence of the series ∑ (1 / 3^n), n=1 to ∞. Calculate the limit of the nth root of |a_n| as n approaches infinity.

Use the Root Test to determine if the infinite series ∑ (1 / 3^n), from n=1 to ∞, converges or diverges. This problem explores the application of the Root Test, including limits and the convergence criteria. Suitable for university-level calculus students.

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