### Problem StatementConsider the following series:\[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln(2) \]#### (a) Use the Alternating Series Remainder Theorem to determine the smallest number of terms required to approximate the sum of the convergent series with an error of less than 0.001.\[ \boxed{} \]#### (b) Use a graphing utility to approximate the sum of the series with an error of less than 0.001. (Round your answer to three decimal places.)\[ \boxed{} \]### ExplanationThe given series is an example of an alternating series. The Alternating Series Remainder Theorem (Leibniz Theorem) can be used to determine the error in approximating the sum of the series with a finite number of terms.- **Alternating Series Remainder Theorem**: For an alternating series $\sum (-1)^n a_n$ where $a_n > 0$, the absolute error $|S - S_n|$ is less than or equal to the first omitted term $a_{n+1}$.#### Requirements:1. **Smallest number of terms for error < 0.001**: Apply the theorem to find the minimum value of $n$ such that: \[ a_{n+1} = \frac{1}{n+1} < 0.001 \]2. **Graphical Approximation**: Use a graphing calculator or software to visually determine the sum and verify the error is less than 0.001.When presenting information related to series and their sums, both analytical and graphical methods provide a comprehensive understanding of convergence and error estimation.

We are given the alternating series with the sum,…

Consider the following. ř (-1)" + ! = In(2) %3D n = 1 (a) Use the Alternating Series Remainder Theorem to determine the smallest number of terms required to approximate the sum of the convergent series with an error of less than 0.001. (b) Use a graphing utility to approximate the sum of the series with an error of less than 0.001. (Round your answer to three decimal places.)

Consider the following. ř (-1)" + ! = In(2) %3D n = 1 (a) Use the Alternating Series Remainder Theorem to determine the smallest number of terms required to approximate the sum of the convergent series with an error of less than 0.001. (b) Use a graphing utility to approximate the sum of the series with an error of less than 0.001. (Round your answer to three decimal places.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Problem Statement

Consider the following series:

\[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln(2) \]

#### (a) Use the Alternating Series Remainder Theorem to determine the smallest number of terms required to approximate the sum of the convergent series with an error of less than 0.001.

\[ \boxed{} \]

#### (b) Use a graphing utility to approximate the sum of the series with an error of less than 0.001. (Round your answer to three decimal places.)

\[ \boxed{} \]

### Explanation

The given series is an example of an alternating series. The Alternating Series Remainder Theorem (Leibniz Theorem) can be used to determine the error in approximating the sum of the series with a finite number of terms.

- **Alternating Series Remainder Theorem**: For an alternating series $\sum (-1)^n a_n$ where $a_n > 0$, the absolute error $|S - S_n|$ is less than or equal to the first omitted term $a_{n+1}$.

#### Requirements:

1. **Smallest number of terms for error < 0.001**: Apply the theorem to find the minimum value of $n$ such that:
\[
a_{n+1} = \frac{1}{n+1} < 0.001
\]

2. **Graphical Approximation**: Use a graphing calculator or software to visually determine the sum and verify the error is less than 0.001.

When presenting information related to series and their sums, both analytical and graphical methods provide a comprehensive understanding of convergence and error estimation.

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