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40-43 Find the radius of convergence and interval of conver- gence of the series. (-1)" Σ (- n=1 40. Σ xn n²5n ∞ 41. Σ n=1 (x + 2)" n4n

40-43 Find the radius of convergence and interval of conver- gence of the series. (-1)" Σ (- n=1 40. Σ xn n²5n ∞ 41. Σ n=1 (x + 2)" n4n

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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### Convergence of Series **40-43** Find the radius of convergence and interval of convergence of the series. #### Problem 40 \[ \sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n^2 5^n} \] #### Problem 41 \[ \sum_{n=1}^{\infty} \frac{(x + 2)^n}{n 4^n} \] --- To solve these problems, we typically use the Ratio Test or Root Test to find the radius of convergence \( R \) and then determine the interval of convergence by testing endpoints. The detailed process involves examining the behavior of the general term of the series as \( n \) approaches infinity.
Transcribed Image Text:### Convergence of Series **40-43** Find the radius of convergence and interval of convergence of the series. #### Problem 40 \[ \sum_{n=1}^{\infty} \frac{(-1)^n x^n}{n^2 5^n} \] #### Problem 41 \[ \sum_{n=1}^{\infty} \frac{(x + 2)^n}{n 4^n} \] --- To solve these problems, we typically use the Ratio Test or Root Test to find the radius of convergence \( R \) and then determine the interval of convergence by testing endpoints. The detailed process involves examining the behavior of the general term of the series as \( n \) approaches infinity.
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