Graph Interpretation and Rough Estimate of the Sum

From the graph:

• The sequence of partial sums $S_n$ (orange squares) converges quickly and stabilizes around a value slightly above 0.10.
• The sequence of terms $a_n$ (orange circles) alternates between positive and negative values, decreasing in magnitude as $n$ increases.

Based on the graph, a rough estimate of the sum of the series is approximately 0.11.

Step 2: Using the Alternating Series Estimation Theorem

The Alternating Series Estimation Theorem states that for an alternating series that meets the following criteria:

1. $a_{n+1} \leq a_n$ for all $n$.
2. $\lim_{n \to \infty} a_n = 0$.

The error in using the $n$th partial sum $S_n$ to estimate the sum of the series is less than or equal to the absolute value of the first omitted term $a_{n+1}$.

Let's calculate the first few terms of the series and use this theorem to estimate the sum correct to four decimal places.### Alternating Series Estimation Theorem Result

• The sum of the series using the first 5 terms (i.e., the partial sum $S_5$) is approximately 0.1226.
• The error estimate for this sum, based on the Alternating Series Estimation Theorem, is about 0.0001.

Therefore, the sum of the series, correct to four decimal places, is 0.1226.

Would you like more details or have any questions?

Related Questions

1. How does the Alternating Series Estimation Theorem guarantee the error bound?
2. Why does the sequence of partial sums converge quickly in this series?
3. How would the convergence change if the denominator were $7^n$ instead of $6^n$?
4. What if the series was not alternating? How would the estimation approach differ?
5. How can we use a computer algorithm to calculate more terms efficiently?
6. Can this series sum be computed using a closed-form expression?
7. How does the rate of decay of $a_n$ affect the convergence speed?
8. What is the significance of the term $n=6$ and beyond in this series?

Tip

For alternating series, the error bound provided by the first omitted term is often very tight, making it a powerful tool for estimation.