Problem 49 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \( n \), \( \operatorname{coth} n \approx 1 \).

Step 3: 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 \( p = 2 \).

Step 4: 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.

Step 5: 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.

Key Concepts

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’s 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.