Problem 46 - Thomas Calculus

Question

Which of the series in Exercises 13 46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \operatorname{sech}^{2} n $$

Step-by-Step Solution

Answer

The series converges based on comparison with a convergent geometric series.

Step 1: Understand the Function

The series given is \( \sum_{n=1}^{\infty} \operatorname{sech}^{2} n \). The function \( \operatorname{sech}(x) \) is defined as the hyperbolic secant, \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \), where \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). Hence, \( \operatorname{sech}^{2}(n) = \left(\frac{2}{e^n + e^{-n}}\right)^2 \).

Step 2: Determine Behavior of Terms

As \( n \to \infty \), the denominator \( e^n + e^{-n} \) behaves approximately like \( e^n \) because \( e^{-n} \) becomes negligible. Therefore, \( \operatorname{sech}^{2}(n) \approx \left(\frac{2}{e^n}\right)^2 = \frac{4}{e^{2n}} \), which decreases exponentially.

Step 3: Apply the Comparison Test

Since each term \( \operatorname{sech}^{2}(n) \approx \frac{4}{e^{2n}} \) is positive and behaves similarly to the negative exponential function \( \frac{4}{e^{2n}} \), compare the given series with the geometric series \( \sum \frac{4}{e^{2n}} \). The geometric series \( \sum r^n \) converges if \( |r| < 1 \). Here ratio \( r = \frac{1}{e^{2}} < 1 \).

Step 4: Conclude Convergence Using Comparison

Since \( \operatorname{sech}^{2}(n) \) is eventually similar to a convergent geometric series \( \sum \frac{4}{e^{2n}} \), by the comparison test, the given series \( \sum_{n=1}^{\infty} \operatorname{sech}^{2} n \) also converges.

Key Concepts

Comparison Test

In the world of series convergence, the Comparison Test is an incredibly useful tool. It helps us determine if a series converges or diverges by comparing it to another series with known behavior.Here's how it works:

• First, you need two series: the one you're investigating and another one with a known convergence pattern.
• The series must have all positive terms for the Comparison Test to be applicable.
• Compare the sizes of the terms between both series.
• If the original series has terms smaller than the comparable known convergent series, then it converges too. Conversely, if it's larger than a known divergent series, it diverges.

This test is elegant because it often simplifies complex series evaluations by referring to known benchmarks.
In our original exercise, we compared the series of \( \sum_{n=1}^{\infty} \operatorname{sech}^{2} n \) with a geometric series \( \sum \frac{4}{e^{2n}} \). This helped confirm its convergence, as both behaved similarly for large values of \( n \). This makes the Comparison Test both powerful and relatively straightforward to apply.

Geometric Series

Geometric series are among the most well-known and easily recognized series. They take the form \( \sum ar^n \) where \( a \) is the first term and \( r \) is the common ratio.Key points to remember about geometric series:

• If the absolute value of the ratio \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series diverges.
• The sum of a convergent geometric series can be calculated using the formula \( a/(1-r) \).

These characteristics make geometric series a benchmark for evaluating others, especially using the Comparison Test.
In the exercise, \( \operatorname{sech}^{2}(n) \approx \frac{4}{e^{2n}} \) resembles a geometric series with a ratio \( r = 1/e^2 < 1 \), which implies convergence. Recognizing these patterns is fundamental for evaluating series.

Hyperbolic Functions

Hyperbolic functions, much like their trigonometric counterparts, are essential in many areas of mathematics. They are defined using exponential functions and have unique properties.For example:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)
• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)
• \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \)
• \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \)

These functions are not only useful in calculus but also in solving differential equations and modeling real-world scenarios.
Our series \( \sum_{n=1}^{\infty} \operatorname{sech}^{2} n \) involves the hyperbolic secant. The function \( \operatorname{sech}(x) \) is defined as the reciprocal of \( \cosh(x) \), enabling us to express series terms based on exponential functions. This perspective was instrumental in simplifying the terms for convergence evaluation.