of values of the variable x for which the following geometric power se vergence) x+9x² +27x³ +81x* +, :-1 (x-1)² (x-1)' , (x-1)* +... 2 + 8 16 4
of values of the variable x for which the following geometric power se vergence) x+9x² +27x³ +81x* +, :-1 (x-1)² (x-1)' , (x-1)* +... 2 + 8 16 4
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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![**Title:** Interval of Convergence for Geometric Power Series
**Introduction:**
In this section, we will explore how to determine the range of values for the variable \( x \) that allows the convergence of a given geometric power series. This range is known as the interval of convergence.
**Problem Statement:**
Find the range of values of the variable \( x \) for which the following geometric power series converges:
**Series:**
**a)**
\[
G(x) = 1 + 3x + 9x^2 + 27x^3 + 81x^4 + \ldots
\]
**b)**
\[
G(x) = 1 - \frac{x-1}{2} + \frac{(x-1)^2}{4} - \frac{(x-1)^3}{8} + \frac{(x-1)^4}{16} + \ldots
\]
**c)**
\[
G(x) = 1 + 0.4(x+3) + 0.16(x+3)^2 + 0.064(x+3)^3 + 0.0256(x+3)^4 + \ldots
\]
**Discussion:**
In each series, identify the common ratio and apply the criteria for convergence of geometric series to find the interval of convergence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7d01a9fa-2d22-41b6-ab89-d6b153d8821e%2Fedb456d9-cf23-417d-b373-b262284a7b83%2Fa9qtvuqd_processed.png&w=3840&q=75)
Transcribed Image Text:**Title:** Interval of Convergence for Geometric Power Series
**Introduction:**
In this section, we will explore how to determine the range of values for the variable \( x \) that allows the convergence of a given geometric power series. This range is known as the interval of convergence.
**Problem Statement:**
Find the range of values of the variable \( x \) for which the following geometric power series converges:
**Series:**
**a)**
\[
G(x) = 1 + 3x + 9x^2 + 27x^3 + 81x^4 + \ldots
\]
**b)**
\[
G(x) = 1 - \frac{x-1}{2} + \frac{(x-1)^2}{4} - \frac{(x-1)^3}{8} + \frac{(x-1)^4}{16} + \ldots
\]
**c)**
\[
G(x) = 1 + 0.4(x+3) + 0.16(x+3)^2 + 0.064(x+3)^3 + 0.0256(x+3)^4 + \ldots
\]
**Discussion:**
In each series, identify the common ratio and apply the criteria for convergence of geometric series to find the interval of convergence.
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