Problem 46 - Thomas Calculus

Question

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \operatorname{csch} n $$

Step-by-Step Solution

Answer

The series converges conditionally; it does not converge absolutely.

Step 1: Identify the Series

The series given is \( \sum_{n=1}^{\infty} (-1)^{n} \operatorname{csch} n \), where \( \operatorname{csch} n = \frac{1}{\sinh n} \). This is an alternating series because the terms include \((-1)^n\).

Step 2: Check for Absolute Convergence

To determine absolute convergence, we must check the convergence of \( \sum_{n=1}^{\infty} \left| (-1)^{n} \operatorname{csch} n \right| = \sum_{n=1}^{\infty} \operatorname{csch} n \). The hyperbolic sine function, \( \sinh n \), grows exponentially as \( n \to \infty \), so \( \operatorname{csch} n = \frac{1}{\sinh n} \) goes to zero but not quickly enough for the absolute series to converge. Thus, the series does not converge absolutely.

Step 3: Use the Alternating Series Test

The alternating series test states that an alternating series \( \sum_{n=1}^{\infty} (-1)^{n} a_n \) converges if \( a_n \) decreases monotonically to 0 as \( n \to \infty \). Here, \( a_n = \operatorname{csch} n \) is positive and decreases since \( \operatorname{csch} n \to 0 \) as \( n \to \infty \), meeting the test's conditions. Therefore, \( \sum_{n=1}^{\infty} (-1)^{n} \operatorname{csch} n \) converges.

Step 4: Conclusion on Series Behavior

Since the series \( \sum_{n=1}^{\infty} (-1)^{n} \operatorname{csch} n \) does not converge absolutely but does converge conditionally, we can conclude that the series converges by the alternating series test but not absolutely.

Key Concepts

Absolute Convergence

Absolute convergence is a fascinating concept in the study of series. A series \( \sum_{n=1}^{\infty} a_n \) is said to converge absolutely if the series of its absolute values, \( \sum_{n=1}^{\infty} |a_n| \), also converges. This conditions the series to behave correctly under term rearrangement and guarantees convergence.
In the exercise solution, the series in question is \( \sum_{n=1}^{\infty} (-1)^{n} \operatorname{csch} n \). To test for absolute convergence, we take the series of the absolute values: \( \sum_{n=1}^{\infty} \left| (-1)^{n} \operatorname{csch} n \right| = \sum_{n=1}^{\infty} \operatorname{csch} n \).
Here, \( \operatorname{csch} n \) is the hyperbolic cosecant function, which shrinks to zero, but not quickly enough for the absolute series to converge. The function shrinks as the hyperbolic sine, so even though it approaches zero, it does so slower than required, resulting in a divergence in terms of absolute convergence.
Key Points:

• An absolutely convergent series is one where the absolute values of terms converge.
• In the case of \( \operatorname{csch} n \), it goes to zero but not rapidly enough, causing the series not to converge absolutely.

Alternating Series Test

The Alternating Series Test is a reliable tool for checking the convergence of series with alternating signs. A classic alternating series looks like \( \sum_{n=1}^{\infty} (-1)^n a_n \).
The test affirms that such a series converges if the absolute value of the terms \( a_n \) is:

• Monotonically decreasing, meaning each term is smaller than or equal to the previous term, and
• Approaching zero as \( n \to \infty \).

In the provided example, the series \( \sum_{n=1}^{\infty} (-1)^{n} \operatorname{csch} n \) fits the structure of an alternating series. The term \( a_n = \operatorname{csch} n \) decreases monotonically as \( n \to \infty \) because the values approach zero. Thus, it meets the test's criteria, indicating the series converges conditionally using this test.
Significance:

• Not every convergent series is absolutely convergent. Conditional convergence is an important concept to grasp.
• For alternating series, the alternation and gradual shrinkage to zero fosters convergence.

Hyperbolic Functions

Hyperbolic functions often parallel trigonometric functions but relate to hyperbolas instead of circles. Common hyperbolic functions include \( \sinh x \), \( \cosh x \), and more, each defined as exponential function expressions.
The exercise involves the hyperbolic cosecant function, defined as \( \operatorname{csch} x = \frac{1}{\sinh x} \). Key properties of hyperbolic functions to understand are:

• The functions grow exponentially with increasing values of \( x \).
• They share relationships analogous to trigonometric identities.

In the scenario of the given series, since \( \sinh n \) grows exponentially, \( \operatorname{csch} n \) tends towards zero. Yet, due to its slower decay, the series does not achieve absolute convergence.
Essentials to Remember:

• Hyperbolic functions, while resembling trigonometric functions, deal with hyperbolas and have distinct growth behaviors.
• The understanding of exponential behavior helps interpret series behavior, especially in convergence examination.