Domanda del libro di testo
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 10, Problema 10.7.65f
Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
f. Diverges for x = 4.9
Guida verificata passo dopo passo1
Identify the center of the power series, which is at \(x = 2\), since the series is given as \(\sum a_n (x - 2)^n\).
Use the information about convergence and divergence at specific points to find the radius of convergence \(R\). The series converges at \(x = -1\) and diverges at \(x = 6\).
Calculate the distance from the center to these points: \(|-1 - 2| = 3\) and \(|6 - 2| = 4\). Since the series converges at \(x = -1\) (distance 3) and diverges at \(x = 6\) (distance 4), the radius of convergence \(R\) satisfies \(3 \leq R < 4\).
Determine whether the series converges or diverges at \(x = 4.9\) by calculating the distance from the center: \(|4.9 - 2| = 2.9\).
Since \(2.9 < 3 \leq R\), the point \(x = 4.9\) lies within the radius of convergence, so the series converges at \(x = 4.9\). Therefore, the statement that it diverges at \(x = 4.9\) is false.
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Radius of Convergence
The radius of convergence is the distance from the center of a power series within which the series converges absolutely. For a series centered at x = 2, it converges for all x such that |x - 2| < R, where R is the radius. Knowing convergence and divergence at specific points helps estimate R.
Video consigliato:
07:36
Radius of Convergence
Interval of Convergence
The interval of convergence is the set of all x-values for which the power series converges. It is centered at the series' center (here, x = 2) and extends R units in both directions. Convergence at boundary points must be checked separately, as behavior can differ.
Video consigliato:
08:44Interval of Convergence
Testing Convergence at Specific Points
To determine if a series converges or diverges at a particular x, compare the distance |x - center| to the radius of convergence. If the distance is less than R, the series converges; if greater, it diverges. If equal, convergence must be tested individually.
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Percorso guidato
07:51Choosing a Convergence Test
Pratica correlata
Domanda del libro di testo
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]
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Domanda del libro di testo
Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
e. Diverges for x = 8
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