6. S= 00 (-1)^2n 3" n=0 =o =0--+ 2 4 39 6 8 + 27 81 (a) Show steps and find lim antl 11-0 an +... (b) Based on (a), does S converge or diverge? Does T = Σ n=0 2n 3" converge or diverge?
6. S= 00 (-1)^2n 3" n=0 =o =0--+ 2 4 39 6 8 + 27 81 (a) Show steps and find lim antl 11-0 an +... (b) Based on (a), does S converge or diverge? Does T = Σ n=0 2n 3" converge or diverge?
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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![**Problem 6:**
Given the series:
\[ S = \sum_{n=0}^{\infty} \frac{(-1)^n 2n}{3^n} = 0 - \frac{2}{3} + \frac{4}{9} - \frac{6}{27} + \frac{8}{81} + \ldots \]
**Tasks:**
(a) **Show steps and find** \( \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| \).
(b) **Based on (a), does \( S \) converge or diverge? Does**
\[ T = \sum_{n=0}^{\infty} \frac{2n}{3^n} \]
**converge or diverge?**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14b65783-eb23-4822-951c-32d4f7635fee%2F75f07c66-0d9d-43b2-b001-7492a8459fc2%2Fmmnmn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6:**
Given the series:
\[ S = \sum_{n=0}^{\infty} \frac{(-1)^n 2n}{3^n} = 0 - \frac{2}{3} + \frac{4}{9} - \frac{6}{27} + \frac{8}{81} + \ldots \]
**Tasks:**
(a) **Show steps and find** \( \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| \).
(b) **Based on (a), does \( S \) converge or diverge? Does**
\[ T = \sum_{n=0}^{\infty} \frac{2n}{3^n} \]
**converge or diverge?**
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