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Consider the power series (-1)n Σ F(x) = n25n n=1 We are interested in the domain of the function F(x) 1. What is the center of the power series? 2. Use the Ratio Test to show that the power series converges for -5< x < 5 3. Determine if the power series F(r) converges or diverges for r-5. Include a complete argument 4. Determine if the power series F(r) converges or diverges for r 5. Include a complete argument 5. For what values of r does the power series diverge? (Hint: Reference your work with the Ratio Test.) 6. What is the domain of the function F(x)? The domain of a power series is always an interval that is centered at the center of the power series. Thus, the domain of a power series is referred to as the Interval of Convergence of the power series. The distance from the center to the exterior of the Interval of Convergence is called the Radius of Convergence. 7. What is the Radius of Convergence for F(r)? Note that this could have been answered immediately from your work with the Ratio Test in #2. 8. Sketch a pieture of the Interval of Convergence for F(r) similar to the one in Figure 3 in §11.8. Label the center of the power series, ther-values for which the power series converges, and the r-values for which the power series diverges

Consider the power series (-1)n Σ F(x) = n25n n=1 We are interested in the domain of the function F(x) 1. What is the center of the power series? 2. Use the Ratio Test to show that the power series converges for -5< x < 5 3. Determine if the power series F(r) converges or diverges for r-5. Include a complete argument 4. Determine if the power series F(r) converges or diverges for r 5. Include a complete argument 5. For what values of r does the power series diverge? (Hint: Reference your work with the Ratio Test.) 6. What is the domain of the function F(x)? The domain of a power series is always an interval that is centered at the center of the power series. Thus, the domain of a power series is referred to as the Interval of Convergence of the power series. The distance from the center to the exterior of the Interval of Convergence is called the Radius of Convergence. 7. What is the Radius of Convergence for F(r)? Note that this could have been answered immediately from your work with the Ratio Test in #2. 8. Sketch a pieture of the Interval of Convergence for F(r) similar to the one in Figure 3 in §11.8. Label the center of the power series, ther-values for which the power series converges, and the r-values for which the power series diverges

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 50E
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Consider the power series (-1)n Σ F(x) = n25n n=1 We are interested in the domain of the function F(x) 1. What is the center of the power series? 2. Use the Ratio Test to show that the power series converges for -5< x < 5 3. Determine if the power series F(r) converges or diverges for r-5. Include a complete argument
Transcribed Image Text:Consider the power series (-1)n Σ F(x) = n25n n=1 We are interested in the domain of the function F(x) 1. What is the center of the power series? 2. Use the Ratio Test to show that the power series converges for -5< x < 5 3. Determine if the power series F(r) converges or diverges for r-5. Include a complete argument
4. Determine if the power series F(r) converges or diverges for r 5. Include a complete argument 5. For what values of r does the power series diverge? (Hint: Reference your work with the Ratio Test.) 6. What is the domain of the function F(x)? The domain of a power series is always an interval that is centered at the center of the power series. Thus, the domain of a power series is referred to as the Interval of Convergence of the power series. The distance from the center to the exterior of the Interval of Convergence is called the Radius of Convergence. 7. What is the Radius of Convergence for F(r)? Note that this could have been answered immediately from your work with the Ratio Test in #2. 8. Sketch a pieture of the Interval of Convergence for F(r) similar to the one in Figure 3 in §11.8. Label the center of the power series, ther-values for which the power series converges, and the r-values for which the power series diverges
Transcribed Image Text:4. Determine if the power series F(r) converges or diverges for r 5. Include a complete argument 5. For what values of r does the power series diverge? (Hint: Reference your work with the Ratio Test.) 6. What is the domain of the function F(x)? The domain of a power series is always an interval that is centered at the center of the power series. Thus, the domain of a power series is referred to as the Interval of Convergence of the power series. The distance from the center to the exterior of the Interval of Convergence is called the Radius of Convergence. 7. What is the Radius of Convergence for F(r)? Note that this could have been answered immediately from your work with the Ratio Test in #2. 8. Sketch a pieture of the Interval of Convergence for F(r) similar to the one in Figure 3 in §11.8. Label the center of the power series, ther-values for which the power series converges, and the r-values for which the power series diverges
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