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

검증된 단계별 안내

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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주요 개념

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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.

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.

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.

Power SeriesIn 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