Domanda del libro di testo
Convergent Series
Find the sums of the series in Exercises 19–24.
∑ (from n = 2 to ∞) -2/[n(n+1)]
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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.PE.55
Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) (csch n)xⁿ
Guida verificata passo dopo passo1
Identify the given power series: \( \sum_{n=1}^{\infty} \text{csch}(n) x^n \), where \( \text{csch}(n) = \frac{1}{\sinh(n)} \).
To find the radius of convergence, use the Root Test or Ratio Test. Here, the Root Test is convenient: compute \( \lim_{n \to \infty} \sqrt[n]{|\text{csch}(n) x^n|} = \lim_{n \to \infty} |x| \sqrt[n]{|\text{csch}(n)|} \).
Analyze the behavior of \( \text{csch}(n) \) as \( n \to \infty \). Since \( \sinh(n) \) grows exponentially, \( \text{csch}(n) \) tends to zero exponentially, so \( \sqrt[n]{|\text{csch}(n)|} \) tends to a limit less than 1.
Use this limit to find the radius of convergence \( R \) by solving \( \lim_{n \to \infty} \sqrt[n]{|\text{csch}(n)|} \cdot |x| < 1 \), which simplifies to \( |x| < R \).
For the interval of convergence, check the endpoints \( x = \pm R \) by substituting into the series and testing for convergence using appropriate tests (e.g., Alternating Series Test or Comparison Test). Then determine where the series converges absolutely (when \( \sum |\text{csch}(n) x^n| \) converges) and conditionally (when the original series converges but not absolutely).
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Power Series and Radius of Convergence
A power series is an infinite sum of terms in the form a_n(x - c)^n, where a_n are coefficients and c is the center. The radius of convergence is the distance from c within which the series converges. It can be found using tests like the Ratio or Root Test, determining where the series behaves well.
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07:36
Radius of Convergence
Absolute and Conditional Convergence
A series converges absolutely if the series of absolute values converges, ensuring strong convergence regardless of term signs. Conditional convergence occurs when the original series converges but the absolute value series diverges, often involving alternating or sign-changing terms.
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Percorso guidato
07:51Choosing a Convergence Test
Hyperbolic Cosecant Function (csch n) in Series Terms
The hyperbolic cosecant, csch n, is defined as 1/sinh n and decreases exponentially as n increases. Understanding its behavior helps analyze the coefficients' size in the series, which affects convergence tests and the radius of convergence for the power series.
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07:32Representing Functions as Power Series
Pratica correlata
Domanda del libro di testo
Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) xⁿ/nⁿ
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Domanda del libro di testo
The series
sec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯
converges to sec x for −π/2 < x < π/2.
a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?
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Domanda del libro di testo
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 1 + (0.9)ⁿ
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Domanda del libro di testo
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (-4)ⁿ/n!
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Domanda del libro di testo
Maclaurin Series
Find Taylor series at x = 0 for the functions in Exercises 63–70.
cos (x³/√5)
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