Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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. ###
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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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