91Ó°ÊÓ

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola instead of a circle. They include hyperbolic sine (\text{sinh}), hyperbolic cosine (\text{cosh}), and the hyperbolic secant (\text{sech}). For this exercise, we focus on the hyperbolic secant, defined as \(\text{sech}(x) = \frac{2}{e^x + e^{-x}}\).

Just like trigonometric functions, hyperbolic functions have various useful properties. For instance, \(\text{sech}(x)\) tends to zero as \(x\) increases, because \(e^x\) grows exponentially while \(e^{-x}\) shrinks.

Understanding hyperbolic functions is crucial in the given series \(\text{\text{sech}}^2(n)\). It's important to recognize that this function decays exponentially, meaning it gets very small quickly as \(n\) becomes large.

The Comparison Test is a powerful tool in determining the convergence or divergence of series. Here’s how it works: If you can compare your given series to another series whose behavior is known, you can infer the behavior of your series.

The steps for applying the Comparison Test are straightforward:

• Identify a series that is easy to understand and whose convergence or divergence is already known.
• Compare the terms of your series to the terms of the known series.
• If the terms of your series are always smaller than (or larger than) a known convergent (or divergent) series and meet certain criteria, the test can confirm the behavior of your series.

In the given exercise, \(\text{\text{sech}}^2(n)\) is compared with \(e^{-2n}\). Since \(e^{-2n}\) is a geometric series with a ratio less than 1, it converges.

By comparing \(\text{\text{sech}}^2(n)\) to this convergent geometric series, we infer that \(\text{\text{sech}}^2(n)\) also converges because \(\text{\text{sech}}^2(n)\) decays even faster than \(e^{-2n}\).

A geometric series is one of the simplest types of series and is defined as the sum of the terms of a geometric sequence. A geometric sequence is where each term is a constant multiple of the previous term.

A geometric series can be written as \(\text{\text{sum}} = ar^0 + ar^1 + ar^2 + \text{...}\) where \(a\) is the first term and \(r\) is the common ratio.

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

In the given problem, \(e^{-2n}\) forms a geometric series with \(a = 1\) and \(r = e^{-2}\), which is less than 1. This means the series converges.

By leveraging the properties of the geometric series, we provide a benchmark to help compare and conclude the behavior of \(\text{\text{sech}}^2(n)\) effectively.