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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![### Question: Convergence of Geometric Series
For which values of \( x \) does the geometric series
\[ \sum_{n=0}^{\infty} 2^n \, x^n \]
converge?
#### Select one:
- **a.** \( x \leq 1 \)
- **b.** \( |2x| > 3 \)
- **c.** \( x > 1 \)
- **d.** \( |x| < \frac{1}{2} \)
---
#### Explanation:
To determine the convergence of the geometric series \(\sum_{n=0}^{\infty} 2^n \, x^n \), consider the general form of a geometric series:
\[ \sum_{n=0}^{\infty} ar^n \]
where \( a \) is the first term and \( r \) is the common ratio.
For the series to converge, the common ratio \( r \) must satisfy \( |r| < 1 \).
In the given series, \( r = 2x \).
Therefore, we need:
\[ |2x| < 1 \]
This simplifies to:
\[ |x| < \frac{1}{2} \]
So, the correct answer is:
- **d.** \( |x| < \frac{1}{2} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd8f2877-2578-48c0-ab77-5dda294aaf93%2F0873a954-5ceb-4ee2-a1ce-03ffce3c56fa%2F4svi0m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question: Convergence of Geometric Series
For which values of \( x \) does the geometric series
\[ \sum_{n=0}^{\infty} 2^n \, x^n \]
converge?
#### Select one:
- **a.** \( x \leq 1 \)
- **b.** \( |2x| > 3 \)
- **c.** \( x > 1 \)
- **d.** \( |x| < \frac{1}{2} \)
---
#### Explanation:
To determine the convergence of the geometric series \(\sum_{n=0}^{\infty} 2^n \, x^n \), consider the general form of a geometric series:
\[ \sum_{n=0}^{\infty} ar^n \]
where \( a \) is the first term and \( r \) is the common ratio.
For the series to converge, the common ratio \( r \) must satisfy \( |r| < 1 \).
In the given series, \( r = 2x \).
Therefore, we need:
\[ |2x| < 1 \]
This simplifies to:
\[ |x| < \frac{1}{2} \]
So, the correct answer is:
- **d.** \( |x| < \frac{1}{2} \)
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