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The problem in the image involves the infinite series:

\[
\sum_{n=1}^{\infty} (-1)^{n-1} \frac{n}{6^n}
\]

### Solution:

This is an alternating series, and the general term \(a_n\) is given by:

\[
a_n = \frac{n}{6^n}
\]

### 1. **Check for Convergence:**

To determine if the series converges, we can apply the **Alternating Series Test** (Leibniz criterion). The series \(\sum (-1)^{n-1} a_n\) will converge if:

- The sequence \(a_n = \frac{n}{6^n}\) is decreasing.
- \(\lim_{n \to \infty} a_n = 0\).

Let's check these conditions:

1. **Monotonic Decreasing:**
   - We need to check if \(a_{n+1} < a_n\) for all \(n\).
   \[
   a_{n+1} = \frac{n+1}{6^{n+1}}, \quad a_n = \frac{n}{6^n}
   \]
   We compare the ratio:
   \[
   \frac{a_{n+1}}{a_n} = \frac{\frac{n+1}{6^{n+1}}}{\frac{n}{6^n}} = \frac{n+1}{6n}
   \]
   Since \(\frac{n+1}{6n} < 1\) for all \(n\), \(a_n\) is indeed decreasing.

2. **Limit to Zero:**
   \[
   \lim_{n \to \infty} \frac{n}{6^n} = 0
   \]
   As \(n\) grows, the exponential \(6^n\) grows much faster than \(n\), so the limit is zero.

Both conditions are satisfied, so the series converges.

### 2. **Estimate the Sum using the Alternating Series Estimation Theorem:**

The **Alternating Series Estimation Theorem** states that the error \(R_n\) when approximating the sum \(S\) of the infinite series by the \(n\)-th partial sum \(S_n\) is less than the magnitude of the first omitted term \(a_{n+1}\).

So, if we want to estimate the sum up to \(n\) terms, the error is less than \(a_{n+1} = \frac{n+1}{6^{n+1}}\).

### 3. **Graphing the Sequence and the Partial Sums:**

The problem asks to graph the sequence of terms \(a_n = \frac{n}{6^n}\) and the partial sums \(S_n\). These graphs would typically show:

- The sequence \(a_n\) decreasing rapidly towards zero.
- The partial sums \(S_n\) converging to the actual sum of the series.

To perform these calculations and graph the results, you'd typically use a software tool or graphing calculator. You could also plot the first few terms manually.

### Final Answer:

- **Convergence:** The series converges.
- **Sum Estimation:** Use the Alternating Series Estimation Theorem to approximate the sum \(S\) by calculating partial sums and ensuring the error \(R_n\) is less than \(a_{n+1}\).

---

Would you like to see the partial sums calculated for specific terms or need more details on the estimation process?

Here are some related questions:

1. What is the Alternating Series Test?
2. How do you determine if a series is monotonic?
3. Why does \( \lim_{n \to \infty} \frac{n}{6^n} = 0 \)?
4. How does the Alternating Series Estimation Theorem work?
5. What is the significance of the first omitted term in series estimation?
6. How do you plot partial sums for a series?
7. What is the difference between absolute and conditional convergence?
8. How do exponential functions compare to polynomial growth?

**Tip:** When estimating the sum of an alternating series, remember that the error is always less than the first omitted term, which can greatly simplify calculations!

Learn how to solve the infinite series problem involving (-1)^{n-1} * n / 6^n step-by-step. This problem covers the application of the Alternating Series Test, convergence criteria, and the Alternating Series Estimation Theorem. Suitable for advanced undergraduate students and higher levels of study.

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