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 2: Applying the Ratio Test to Determine Convergence or Divergence of Series**
**Objective:**
Use the ratio test to determine whether the following series are convergent or divergent.
**Series (a):**
\[
\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \frac{1}{2^5} + \cdots
\]
**Series (b):**
\[
3 + \frac{3^2}{2} + \frac{3^3}{3} + \frac{3^4}{4} + \frac{3^5}{5} + \cdots
\]
**Instructions:**
1. Apply the ratio test, which involves calculating the limit:
\[
L = \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right|
\]
2. Interpret the result:
- If \( L < 1 \), the series is convergent.
- If \( L > 1 \) or \( L\) is infinite, the series is divergent.
- If \( L = 1 \), the test is inconclusive.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3e671be-5699-41c7-ad14-09ef8165fba8%2Fa0ec37e2-52ac-4fd6-9058-261b4366deb3%2Fgmmkfb6_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2: Applying the Ratio Test to Determine Convergence or Divergence of Series**
**Objective:**
Use the ratio test to determine whether the following series are convergent or divergent.
**Series (a):**
\[
\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \frac{1}{2^5} + \cdots
\]
**Series (b):**
\[
3 + \frac{3^2}{2} + \frac{3^3}{3} + \frac{3^4}{4} + \frac{3^5}{5} + \cdots
\]
**Instructions:**
1. Apply the ratio test, which involves calculating the limit:
\[
L = \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right|
\]
2. Interpret the result:
- If \( L < 1 \), the series is convergent.
- If \( L > 1 \) or \( L\) is infinite, the series is divergent.
- If \( L = 1 \), the test is inconclusive.
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