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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve the sub-parts 3 till 6.
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
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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