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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![### Infinite Integral Convergence Analysis
**Objective:** Determine whether the following integral converges.
\[ \int_{1}^{\infty} \frac{7}{\sqrt[7]{x}} \, dx \]
**Procedure to Determine Convergence:**
1. **Identify the integral's form**:
\[ \int_{1}^{\infty} f(x) \, dx \]
2. **Function Analysis**:
\[ f(x) = \frac{7}{\sqrt[7]{x}} = 7 x^{-\frac{1}{7}} \]
3. **Integral Convergence Test**:
\[ \int_{1}^{\infty} 7 x^{-\frac{1}{7}} \, dx \]
4. **Integral Calculation**:
- **Determine the antiderivative of \(7 x^{-\frac{1}{7}}\)**:
\[ \int 7 x^{-\frac{1}{7}} \, dx = 7 \cdot \frac{x^{1 - \frac{1}{7}}}{1 - \frac{1}{7}} = 7 \cdot \frac{x^{\frac{6}{7}}}{\frac{6}{7}} = \frac{49}{6} x^{\frac{6}{7}} \]
- **Apply the limits**:
\[ \left[\frac{49}{6} x^{\frac{6}{7}} \right]_{1}^{\infty} \]
- Evaluate at the lower limit:
\[ \left[\frac{49}{6} x^{\frac{6}{7}} \right]_{x=1} = \frac{49}{6} \]
- Evaluate at the upper limit:
\[ \lim_{x \to \infty} \frac{49}{6} x^{\frac{6}{7}} \]
Since \( \frac{6}{7} > 0 \), this term diverges to \(\infty\).
5. **Conclusion**:
- As \( x \to \infty \), \(\frac{49}{6} x^{\frac{6}{7}} \to \infty\).
- Therefore, the integral diverges.
**Result**: The given integral does not converge.
###](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F4a6d40f9-b44f-4139-bfa1-c85677433ec2%2Fi2t4ocq_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Infinite Integral Convergence Analysis
**Objective:** Determine whether the following integral converges.
\[ \int_{1}^{\infty} \frac{7}{\sqrt[7]{x}} \, dx \]
**Procedure to Determine Convergence:**
1. **Identify the integral's form**:
\[ \int_{1}^{\infty} f(x) \, dx \]
2. **Function Analysis**:
\[ f(x) = \frac{7}{\sqrt[7]{x}} = 7 x^{-\frac{1}{7}} \]
3. **Integral Convergence Test**:
\[ \int_{1}^{\infty} 7 x^{-\frac{1}{7}} \, dx \]
4. **Integral Calculation**:
- **Determine the antiderivative of \(7 x^{-\frac{1}{7}}\)**:
\[ \int 7 x^{-\frac{1}{7}} \, dx = 7 \cdot \frac{x^{1 - \frac{1}{7}}}{1 - \frac{1}{7}} = 7 \cdot \frac{x^{\frac{6}{7}}}{\frac{6}{7}} = \frac{49}{6} x^{\frac{6}{7}} \]
- **Apply the limits**:
\[ \left[\frac{49}{6} x^{\frac{6}{7}} \right]_{1}^{\infty} \]
- Evaluate at the lower limit:
\[ \left[\frac{49}{6} x^{\frac{6}{7}} \right]_{x=1} = \frac{49}{6} \]
- Evaluate at the upper limit:
\[ \lim_{x \to \infty} \frac{49}{6} x^{\frac{6}{7}} \]
Since \( \frac{6}{7} > 0 \), this term diverges to \(\infty\).
5. **Conclusion**:
- As \( x \to \infty \), \(\frac{49}{6} x^{\frac{6}{7}} \to \infty\).
- Therefore, the integral diverges.
**Result**: The given integral does not converge.
###
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