### Alternating Series Remainder Theorem **Problem Statement:** Use the Alternating Series Remainder Theorem to determine the smallest number of terms required to approximate the sum of the series with an error of less than 0.001. **Series:** \[ \sum_{{n=1}}^{\infty} \frac{{(-1)^{n+1}}}{{n^3}} \] **Explanation of the Series:** - This is an alternating series, which means the signs of the terms alternate between positive and negative. This is indicated by \((-1)^{n+1}\). - Each term in the series is given by the formula \(\frac{1}{{n^3}}\), with \(n\) being a natural number starting from 1 up to infinity. **Objective:** To find the smallest number of terms \(n\) such that the error in approximating the sum of the infinite series is less than 0.001. **Key Concept:** The Alternating Series Remainder Theorem states that for an alternating series that satisfies the conditions of convergence, the absolute error \(R_n\) is less than or equal to the absolute value of the first omitted term: \[ |R_n| \leq |a_{n+1}| \] Where \(a_n\) is the nth term of the series. Therefore, to achieve an error less than 0.001, we solve: \[ \frac{1}{(n+1)^3} < 0.001 \]

### Alternating Series Remainder Theorem Problem Statement: Use the Alternating Series Remainder Theorem to determine the smallest number of terms required to approximate the sum of the series with an error of less than 0.001. Series: \[ \sum_{{n=1}}^{\infty} \frac{{(-1)^{n+1}}}{{n^3}} \] Explanation of the Series: - This is an alternating series, which means the signs of the terms alternate between positive and negative. This is indicated by \((-1)^{n+1}\). - Each term in the series is given by the formula \(\frac{1}{{n^3}}\), with \(n\) being a natural number starting from 1 up to infinity. Objective: To find the smallest number of terms \(n\) such that the error in approximating the sum of the infinite series is less than 0.001. Key Concept: The Alternating Series Remainder Theorem states that for an alternating series that satisfies the conditions of convergence, the absolute error \(R_n\) is less than or equal to the absolute value of the first omitted term: \[ |R_n| \leq |a_{n+1}| \] Where \(a_n\) is the nth term of the series. Therefore, to achieve an error less than 0.001, we solve: \[ \frac{1}{(n+1)^3} < 0.001 \]

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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