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The radius of convergence for the power series - xnn² 272 is

The radius of convergence for the power series - xnn² 272 is

Calculus: Early Transcendentals
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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### Mathematics Practice Problems --- #### Problem 13 **Question:** The radius of convergence for the power series \(\sum_{n=1}^{\infty} \frac{x^n n^2}{2^n}\) is **Explanation:** To determine the radius of convergence for the given power series \(\sum_{n=1}^{\infty} \frac{x^n n^2}{2^n}\), you would typically use the ratio test or the root test. These tests help in establishing the interval within which the series converges. --- #### Problem 14 **Question:** Find an estimate of the maximum error in using the first six terms of the alternating series \(\sum_{n=1}^{\infty} \frac{(-2)^n}{n \cdot 3^n}\) to approximate the value of the series. (Round your answer to 5 decimal places.) **Explanation:** To estimate the maximum error when approximating the sum of an alternating series \(\sum_{n=1}^{\infty} \frac{(-2)^n}{n \cdot 3^n} \) using only the first six terms, you apply the alternating series remainder theorem. This theorem states that the absolute error of the approximation is less than or equal to the absolute value of the next term in the series not included in the sum. --- These problems are designed to test your understanding of series convergence and error estimation in infinite series. Ensure you understand the techniques of finding the radius of convergence and estimating errors in alternating series to solve these efficiently.
Transcribed Image Text:### Mathematics Practice Problems --- #### Problem 13 **Question:** The radius of convergence for the power series \(\sum_{n=1}^{\infty} \frac{x^n n^2}{2^n}\) is **Explanation:** To determine the radius of convergence for the given power series \(\sum_{n=1}^{\infty} \frac{x^n n^2}{2^n}\), you would typically use the ratio test or the root test. These tests help in establishing the interval within which the series converges. --- #### Problem 14 **Question:** Find an estimate of the maximum error in using the first six terms of the alternating series \(\sum_{n=1}^{\infty} \frac{(-2)^n}{n \cdot 3^n}\) to approximate the value of the series. (Round your answer to 5 decimal places.) **Explanation:** To estimate the maximum error when approximating the sum of an alternating series \(\sum_{n=1}^{\infty} \frac{(-2)^n}{n \cdot 3^n} \) using only the first six terms, you apply the alternating series remainder theorem. This theorem states that the absolute error of the approximation is less than or equal to the absolute value of the next term in the series not included in the sum. --- These problems are designed to test your understanding of series convergence and error estimation in infinite series. Ensure you understand the techniques of finding the radius of convergence and estimating errors in alternating series to solve these efficiently.
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