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Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places.

### Graph Analysis:
- The **sequence of terms** (yellow) alternates between positive and negative values, converging towards zero as \( n \) increases.
- The **sequence of partial sums** (red) appears to stabilize around a value close to 0.15.

### Rough Estimate:
From the graph, the sum of the series appears to be roughly around **0.15**.

### Alternating Series Estimation Theorem:
To estimate the sum correct to four decimal places using the Alternating Series Estimation Theorem:

\[
\left| S - S_n \right| \leq a_{n+1}
\]

We will calculate \( S_n \) until \( a_{n+1} \) is less than 0.00005 (which ensures four decimal places accuracy). Let's compute that.Using the Alternating Series Estimation Theorem, the sum of the series is approximately **0.1389**, correct to four decimal places.

This estimate was achieved by considering the partial sum \( S_7 \) (i.e., up to \( n = 7 \)), as the term \( a_8 \) becomes sufficiently small (less than 0.00005).

Would you like any further details or have any questions? Here are some related questions you might consider:

1. What does the behavior of the alternating series tell us about its convergence?
2. How would the sum change if the series started from \( n = 0 \) instead of \( n = 1 \)?
3. What is the role of the factor \( (-1)^{n-1} \) in the series?
4. Can the sum be calculated exactly using another method, such as a power series expansion?
5. How does the ratio test apply to this series?
6. What would happen if the denominator was a different base, such as \( 3^n \) instead of \( 5^n \)?
7. How does the rate of convergence compare with other alternating series?
8. How does the series behave if the exponent of \( n \) was different, say \( n^2 \)?

**Tip:** When analyzing alternating series, always consider the magnitude of the terms to determine how quickly the series converges.

Learn how to graph the sequence of terms and partial sums of a series, estimate the series sum using the Alternating Series Estimation Theorem, and achieve accuracy to four decimal places. Explore the convergence behavior and application of estimation theorems in series analysis.

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