91Ó°ÊÓ

The Limit Comparison Test is a powerful tool in calculus used to determine whether a series converges or diverges. The test compares an unknown series to a known benchmark series. If the limit of the ratio of the terms from the two series results in a positive finite number, both series either converge or diverge.
Let's break it down step-by-step:

• Identify: Choose a series that is similar to the one you are analyzing.
• Calculate: Form the limit expression by dividing the term from the unknown series by the term from the known series.
• Simplify and Evaluate: Simplify the expression if necessary and evaluate the limit as the index approaches infinity.
• Result: If the limit is a finite positive number, the behavior of the series (whether it converges or diverges) matches that of the known series.

In our original problem, the series \( \sum_{k=1}^{\infty} \operatorname{sech}^2(k) \) was compared to \( \sum 4e^{-2k} \). Given the limit test result as 1, similar behavior is concluded for both series ensuring convergence.

A geometric series is a fascinating type of series where each term is a constant multiple of the previous one.
It takes the form:\[ a + ar + ar^2 + ar^3 + \ldots \]where \( a \) is the first term and \( r \) is the common ratio.
One crucial point to remember is the fact that a geometric series converges if the absolute value of the common ratio \( |r| \) is less than 1.

• Convergence: An infinite geometric series converges to \( \frac{a}{1-r} \) if \( |r| < 1 \).
• Divergence: If \( |r| \geq 1 \), the series diverges.

In the exercise, the generic terms were approximated by \( 4e^{-2k} \), a geometric series with common ratio \( e^{-2} \, (\text{which is } < 1) \). This indicated convergence of the series, helping use the geometric series as the benchmark for the limit comparison test.

Hyperbolic functions are analogs to the trigonometric functions but involve hyperbolas instead of circles.
The hyperbolic secant, \( \operatorname{sech}(x) \), is especially interesting. It is defined as:\[ \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \]The hyperbolic secant squared, \( \operatorname{sech}^2(x) \), follows naturally as:\[ \operatorname{sech}^2(x) = \left(\frac{2}{e^x + e^{-x}}\right)^2 \]
These functions are smooth and show rapid decay as their argument \( x \) increases. This property makes them useful for series convergence tests, as shown in the exercise. As \( x \to \infty \), \( \operatorname{sech}^2(x) \) behaves similarly to an exponential decay function. This allows the application of methods like the limit comparison test with geometric series to establish convergence.