Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2(-1)^{n+1}}{e^{n}+e^{-n}}=\sum_{n=1}^{\infty}(-1)^{n+1} \operatorname{sech} n $$

Short answer

The series converges because it satisfies the conditions of the Alternating Series Test.

Step-by-step solution

Understanding the Alternating Series Test

The Alternating series test states that an infinite series \(\sum_{n=1}^{\infty}(-1)^{n+1}a_n\) converges if the following two conditions are met: (i) \(a_{n+1} \leq a_n\) for all n and (ii) \(\lim_{n \to \infty} a_n = 0\).

Apply the first condition of the Alternating Series Test

In the given series, the \(a_n\) part is \(\frac{2}{e^{n}+e^{-n}}\). We need to check whether this sequence is decreasing. Algebraically, this is difficult to prove, but graphically or using software, one can see that function \(f(x) = \frac{2}{e^{x}+e^{-x}}\) is decreasing for \(x \geq 1\). So, condition (i) of the Alternating series test is met.

Apply the second condition of the Alternating Series Test

Next, check whether the limit as n approaches infinity of \(a_n\) is zero. Limit as \(n \to \infty\) of \(\frac{2}{e^n + e^{-n}} = 0\), because as n becomes large, e^n becomes huge and \(\frac{2}{e^n+e^{-n}}\) tends towards zero. Thus condition (ii) of the Alternating series test is met.

Conclude the convergence using the Alternating Series Test

Since conditions (i) and (ii) of the Alternating Series Test are satisfied, we can conclude that the given series converges.

Key concepts

Convergence and Divergence of Series

When exploring the vast world of calculus, one key concept students encounter is the convergence and divergence of series. A series is essentially a sum of terms, but when we talk about an infinite series, we're referring to a sum that goes on indefinitely.

The intrigue of such series lies in determining whether their sums approach a finite value as more terms are added—the hallmark of convergence—or if they grow without bound, which signals divergence. The process of figuring out which scenario plays out involves a variety of tests; each designed for specific types of series.

Some such as the Geometric Series Test or the p-Series Test check for patterns or rate of term shrinkage, but for alternating series, which flip signs as they go, the Alternating Series Test is the gold standard. It checks whether the absolute value of terms is decreasing and approaching zero—key indicators of a convergent series.

Alternating Series

An alternating series is one where the signs of the terms alternate between positive and negative. This can lead to a kind of balance where, despite having infinitely many terms, the series doesn't just rocket to infinity or plummet to negative infinity. Instead, it delicately treads a path that may converge to a finite value.

Understanding these series involves recognizing their pattern: typically, a term is multiplied by (-1)^n or (-1)^{n+1} to ensure this alternating behavior. There's an art to it—knowing this alternating character can cause the series' terms to cancel out some of the 'damage' done by previous terms, inching towards a stable sum rather than growing without restraint.

Calculus

Calculus, the branch of mathematics focused on change and motion, is a toolbox brimming with techniques to solve problems involving quantities that are continuously varying. Infinite series, derivatives, and integrals form the backbone of calculus.

In the case of infinite series, calculus provides methods to determine their convergence or divergence—vital for applications in physics, engineering, and beyond. It's this discipline that equips students with tests such as the Alternating Series Test, allowing a meticulous examination of series that may at first glance appear perplexing.

Infinite Series

The term 'infinite series' can evoke a sense of grand scale and boundlessness, often rightfully so. An infinite series is a sum where the number of terms is allowed to grow without limit. Yet, despite this freedom, under certain conditions, these series can converge to finite, comprehendable values.

This dichotomy is what makes infinite series fascinating. From the harmonic series that diverges to the alternating harmonic series that converges, they challenge our intuitions. Wisdom in calculus is thus not merely about handling the finite, but understanding how the infinite can sometimes be just as manageable.

Limits

At the core of calculus is the concept of limits: the idea of approaching a certain value as closely as desired without necessarily reaching it. This concept allows us to handle quantities that are infinitely small or sums of infinitely many terms.

In the context of infinite series, limits help determine the behavior of a series as the number of its terms escalates to infinity. If the series approaches a fixed number, we say the sum converges to that limit. Conversely, if the terms do not inch toward a specific value, the series diverges. Calculating limits, especially for the terms of a series, is thus a pivotal step in discerning the nature of its sum, convergent or divergent.