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Chapter 10: Problem 49

Which of the series in Exercises \(17-56\) converge, and which diverge? Use any method, and give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{\operatorname{coth} n}{n^{2}} $$

Short Answer

Expert verified
The series \( \sum_{n=1}^{\infty} \frac{\operatorname{coth} n}{n^{2}} \) converges.

Step by step solution

01

Recognize the Test Choice

The series given is \( \sum_{n=1}^{\infty} \frac{\operatorname{coth} n}{n^{2}} \). To determine convergence or divergence, we need to choose an appropriate test. The Limit Comparison Test is suitable here because the \( \operatorname{coth} \) function can be challenging to analyze directly, but it behaves similarly to simpler functions at infinity.
02

Analyze the Behavior of Coth n

Recall that \( \operatorname{coth} n = \frac{\cosh n}{\sinh n} \). As \( n \to \infty \), the \( \operatorname{coth} n \to 1 \). Therefore, for large \( n \), \( \operatorname{coth} n \approx 1 \).
03

Choose a Comparison Series

Based on the behavior \( \operatorname{coth} n \approx 1 \) as \( n \to \infty \), we can compare our series with the simpler series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), which is a convergent p-series with \( p = 2 \).
04

Apply the Limit Comparison Test

The Limit Comparison Test requires us to evaluate\[\lim_{n \to \infty} \frac{\frac{\operatorname{coth} n}{n^{2}}}{\frac{1}{n^{2}}} = \lim_{n \to \infty} \operatorname{coth} n.\]Since we know \( \lim_{n \to \infty} \operatorname{coth} n = 1 \), and this limit is a positive finite number, the test indicates that \( \sum_{n=1}^{\infty} \frac{\operatorname{coth} n}{n^{2}} \) converges.
05

State the Conclusion

By the Limit Comparison Test with the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), which is convergent, the series \( \sum_{n=1}^{\infty} \frac{\operatorname{coth} n}{n^{2}} \) also converges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit Comparison Test
The Limit Comparison Test is a powerful tool when evaluating the convergence of series that cannot be directly analyzed. It is particularly useful when you have a complex expression and suspect it behaves similarly to a simpler series. The test works well when you have two positive series \( \sum a_n \) and \( \sum b_n \).
  • You compute \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
  • If this limit is a positive, finite number, both series either converge or diverge together.
In the original problem, \( \operatorname{coth} n \) given in the series behaves similarly to \( 1 \) for large \( n \). Therefore, using the Limit Comparison Test with the p-series \( \sum \frac{1}{n^2} \) allows us to conclude that both series converge.
Hyperbolic Functions
Hyperbolic functions, like \( \operatorname{coth} n \), are counterparts to trigonometric functions but for the hyperbola. The hyperbolic cotangent function \( \operatorname{coth} n \) can be defined as \( \frac{\cosh n}{\sinh n} \) where \( \cosh n = \frac{e^n + e^{-n}}{2} \) and \( \sinh n = \frac{e^n - e^{-n}}{2} \).
  • As \( n \to \infty \), \( \cosh n \) and \( \sinh n \) become dominated by \( e^n \).
  • This results in \( \operatorname{coth} n \to 1 \) for large \( n \).
Understanding hyperbolic functions is important in calculus as they often arise in integration, solving differential equations, and describing real-world phenomena with hyperbolic models.
p-series Convergence
Understanding p-series is key when using comparison tests. A p-series is of the form \( \sum \frac{1}{n^p} \). Their convergence is easy to determine:
  • Converges if \( p > 1 \).
  • Diverges if \( p \leq 1 \).
In the case of the original exercise, the series \( \sum \frac{1}{n^2} \) is relevant. Because \( p = 2 > 1 \), it converges. This makes it a suitable candidate for comparison using the Limit Comparison Test, as determined when analyzing \( \sum \frac{\operatorname{coth} n}{n^2} \). The resemblance allows the analyst to draw accurate conclusions about convergence.
Series Analysis
Analyzing series involves evaluating a sequence of terms added together, to see if they tend to a finite limit as more terms are added. A series can be finite, where we are certain about convergence, or infinite, leading to various tests applied to determine behavior.Here鈥檚 a basic roadmap for such an analysis:
  • Identify the series type or rewrite it in a simpler form.
  • Select and apply a suitable convergence test, like the Comparison Test or Limit Comparison Test.
  • Look at the behavior of terms for large \( n \) to understand the series better.
  • Conclude based on calculated limits and known series types' behaviors.
Guided by these methods, problems such as \( \sum \frac{\operatorname{coth} n}{n^2} \) become more clear, as has been demonstrated in the solution sketch provided.

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Most popular questions from this chapter

Which of the sequences converge, and which diverge? Give reasons for your answers. $$ a_{n}=\frac{2^{n}-1}{3^{n}} $$

Use the identity \(\sin ^{2} x=(1-\cos 2 x) / 2\) to obtain the Maclaurin series for sin \(^{2} x .\) Then differentiate this series to obtain the Maclaurin series for 2 \(\sin x \cos x\) . Check that this is the series for \(\sin 2 x\) .

\begin{equation} \begin{array}{l}{\text { Using a CAS, perform the following steps to aid in answering }} \\ {\text { questions (a) and (b) for the functions and intervals in Exercises }} \\ {57-62 .}\\\\{\text { Step } I : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\ {x=0}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } \int^{(n+1)}(c) \text { associat- }} \\ {\text { ed with the remainder term for each Taylor polynomial. Plot }} \\ {\text { the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approxima- }} \\ {\text { tions together. Discuss the graphs in relation to the informa- }} \\ {\text { tion discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=\frac{x}{x^{2}+1}, \quad|x| \leq 2$$

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=(0.9999)^{n} $$

Which of the sequences converge, and which diverge? Give reasons for your answers. $$ a_{n}=\frac{4^{n+1}+3^{n}}{4^{n}} $$
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