91Ó°ÊÓ

The hyperbolic secant function, symbolized as \( \operatorname{sech}(x) \), derives from hyperbolic trigonometry. It is defined mathematically as \( \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \). This function reflects the inverse of the hyperbolic cosine, \( \cosh(x) \).

It represents how the function diminishes as \( x \) grows. For positive \( x \), the value of \( \operatorname{sech}(x) \) decreases due to the exponential growth of \( e^x \). This functional behavior shows a rapid decline to zero, which plays a critical role in series and convergence analyses.

As the \( x \) increases, the denominator in \( \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \) becomes significantly large, thus making \( \operatorname{sech}(x) \) approach zero. This characteristic is vital when assessing series based on \( \operatorname{sech}(x) \).

In series assessment, determining convergence is crucial for understanding whether a series sums to a finite number.

Several tests help us analyze this:

• The nth-term test - It states that if the limit of the sequence doesn't converge to zero, the series cannot converge.
• The comparison test - It involves comparing a series with another series that is known to converge or diverge.

In our exercise, applying these tests, especially the nth-term test, helps us conclude the series behavior. The test initially checks if the limit of terms \( \operatorname{sech}^2(k) \) is zero, which it is. However, passing this test isn't enough for convergence proof. Hence, moving to a comparison test can be effective, comparing with a simpler geometric series that we know converges.

Hyperbolic functions, like \( \operatorname{sech}(x) \), are analogous to trigonometric functions like sine and cosine. They arise out of hyperbolic geometry and calculus.

These functions include:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)
• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)
• \( \tanh(x) = \frac{ ext{sinh}(x)}{ ext{cosh}(x)} \)
• \( \operatorname{sech}(x) \), as described earlier

Each of these functions has properties beneficial in calculus, such as calculating integrals of hyperbolic functions, solving differential equations, and modeling certain physical phenomena. Their behavior, particularly as \( x \) tends towards infinity, closely links them to exponential growth and decay.

Series comparison is a valuable technique in calculus to determine the convergence or divergence of a series by comparing it to another series of known behavior.

Here are steps for a comparison test:

• Choose a series \( \sum b_k \) with known convergence or divergence characteristics.
• If \( 0 \leq a_k \leq b_k \) for all \( k \), and \( \sum b_k \) converges, then \( \sum a_k \) converges.
• If \( a_k \geq b_k \), and \( \sum b_k \) diverges, then \( \sum a_k \) diverges.

In this problem, we compare \( \operatorname{sech}^2(k) \) to an exponential series \( \frac{4}{e^{2k}} \). Since the exponential series converges (geometric) as it resembles \( r < 1 \), it suggests that \( \sum \operatorname{sech}^2(k) \) behaves similarly, supporting convergence.

A geometric series is a series where each term after the first is the product of the previous term and a constant called the common ratio \( r \).

Mathematically, it is expressed as \( a + ar + ar^2 + ar^3 + \ldots \), which simplifies to:
\( \sum_{n=0}^{\infty} ar^n \).
If \(|r| < 1\), the geometric series converges and its sum can be calculated using the formula:
\[S = \frac{a}{1 - r}\]
This particular series is immensely helpful as a benchmark when comparing other less obvious series. In our exercise, recognizing that \( \operatorname{sech}^2(k) \) eventually mirrors a geometric series ensures predictions of convergence are accurate. By comparing to \( \frac{4}{e^{2k}} \), we see a convergent pattern since the effective ratio \( e^{-2} \) is less than 1. Aiding the conclusion, this reliance on geometric tenets strengthens convergence tests in calculus.