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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.PE.55
Chapter 10, Problem 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

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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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07:51
Choosing 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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Representing Functions as Power Series
Related Practice
Textbook Question

Theory and Examples

Suppose that a₁, a₂, a₃, …, aₙ are positive numbers satisfying the following conditions:

i) a₁ ≥ a₂ ≥ a₃ ≥ …;


ii) the series a₂ + a₄ + a₈ + a₁₆ + … diverges.

Show that the series


a₁/1 + a₂/2 + a₃/3 + …


diverges.

1
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Textbook Question

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 ∞) (x + 4)ⁿ/(n3ⁿ)

Textbook Question

Convergent Series

Find the sums of the series in Exercises 19–24.


∑ (from n = 2 to ∞) -2/[n(n+1)]

Textbook Question

Use series to evaluate the limits in Exercises 29–40.

37. lim (x → 0) ln(1 + x²) / (1 - cos(x))

Textbook Question

Applying the Integral Test

Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑ (from n = 1 to ∞) 1 / n⁰·²

1
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Textbook Question

30. b. By differentiating the series in part (a) term by term, show that

Σ(from n=1 to ∞) n / (n + 1)! = 1.