Question

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

Accepted Answer

Identify the given power series: $$ \sum_{n=1}^{\infty} 	ext{csch}(n) x^n $$, where $$ 	ext{csch}(n) = \frac{1}{\sinh(n)} . To $$find the radius of convergence, use the Root Test or Ratio Test. Here, the Root Test is convenient: compute $$ \lim_{n 	o \infty} \sqrt[n]{|	ext{csch}(n) x^n|} = \lim_{n 	o \infty} |x| \sqrt[n]{|	ext{csch}(n)|} . $$Analyze the behavior of $$ 	ext{csch}(n)  as$$ $$ n 	o \infty . $$Since $$ \sinh(n) $$ grows exponentially, $$ 	ext{csch}(n) $$ tends to zero exponentially, so $$ \sqrt[n]{|	ext{csch}(n)|} $$ tends to a limit less than 1. Use this limit to find the radius of convergence $$ R  by $$solving $$ \lim_{n 	o \infty} \sqrt[n]{|	ext{csch}(n)|} \cdot |x| \u003c 1 $$, which simplifies to $$ |x| \u003c R . $$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 $$ \sum |	ext{csch}(n) x^n| $$ converges) and conditionally (when the original series converges but not absolutely).

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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Key Concepts

Power Series and Radius of Convergence

Absolute and Conditional Convergence

Hyperbolic Cosecant Function (csch n) in Series Terms