91Ó°ÊÓ

Hyperbolic functions are analogs of trigonometric functions but for the hyperbola, just like trigonometric functions relate to the circle. They have unique definitions but share some similar properties, such as periodicity and symmetry. Among these functions is the hyperbolic secant, denoted as \( \operatorname{sech} x \). It is important to understand the form of \( \operatorname{sech} x \), which is given by:
\[ \operatorname{sech} x = \frac{2}{e^x + e^{-x}}. \]
This formula shows that the hyperbolic secant is dependent on the exponential function, which grows very fast as \( x \) increases.
By investigating how these functions behave as \( n \to \infty \), we see that the \( e^n \) term dominates over \( e^{-n} \). Thus, the expression simplifies to \( \operatorname{sech} n \approx \frac{2}{e^n} \). This indicates that the value of \( \operatorname{sech} n \) approaches zero rapidly as \( n \) becomes very large.

A geometric series is one where each term is a constant multiple of the previous term. The general form of a geometric series is:
\[ a + ar + ar^2 + ar^3 + \ldots \]
where \( a \) is the first term and \( r \) is the common ratio of the series.
Geometric series have particular rules for convergence:

• If \( |r| < 1 \), the series converges to \( \frac{a}{1-r} \).
• If \( |r| \geq 1 \), the series diverges.

In the context of this problem, the series \( \sum_{n=1}^{\infty} \operatorname{sech} n \) was compared to the geometric series \( \sum_{n=1}^{\infty} \frac{2}{e^n} \), where \( \frac{1}{e} \) is the common ratio, which is less than 1. Therefore, this indicates convergence of the series.

The Limit Comparison Test is a useful method to determine the convergence or divergence of a series by comparing it to another series whose behavior is known.
To apply the test:

• Take a series \( \sum a_n \) and compare it to a known convergent or divergent series \( \sum b_n \).
• Compute the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} = L \).
• If \( L \) is positive and finite, then both series either converge or diverge together.

In the provided solution, \( a_n = \operatorname{sech} n \) and \( b_n = \frac{2}{e^n} \). The calculated limit:
\[ \lim_{n \to \infty} \frac{\operatorname{sech} n}{\frac{2}{e^n}} = 1, \]
establishes that \( \sum \operatorname{sech} n \) converges because it behaves similarly to the known geometric series \( \sum \frac{2}{e^n} \).

Determining whether a series converges or diverges is fundamental in calculus and analysis. Several tests can be applied to assist with this decision.
Commonly used tests include:

• Geometric Series Test: Determines convergence based on the common ratio \( r \).
• Limit Comparison Test: Used when comparing to a series with known behavior.
• Ratio Test: Evaluates the limit of the ratio of consecutive terms.
• Integral Test: Compares a series to an improper integral.

Each test has specific conditions under which it can be applied, but together, they offer a comprehensive toolkit for analyzing series. In this exercise, the Limit Comparison Test was especially helpful in showing that \( \sum \operatorname{sech} n \) converges as it mirrored a well-understood geometric series behavior. Understanding and selecting the right test can simplify the process of testing series for convergence.