Problem 32 - Thomas Calculus

Question

In Exercises \(17-46,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}} $$

Step-by-Step Solution

Answer

The series converges by the Alternating Series Test.

Step 1: Identify the Type of Series

We are given the series \( \sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}. \) This is an alternating series since the terms switch signs due to \((-2)^n\). We can express the general term as \(a_n = \frac{n \ln n}{2^n}\) ignoring the negative sign.

Step 2: Check the Alternating Series Test

The Alternating Series Test states that if the absolute value of the terms \(a_n\) is decreasing and \(\lim_{n \to \infty} a_n = 0\), then the series converges. We need to verify these conditions for \(a_n = \frac{n \ln n}{2^n}\).

Step 3: Verify Decreasing Terms

To check if \( \frac{n \ln n}{2^n} \) is decreasing, consider the derivative of \( f(x) = \frac{x \ln x}{2^x} \) using the quotient rule: \(f'(x) = \frac{2^x (1 + \ln x) - x \ln x (2^x \ln 2)}{(2^x)^2}\). Simplifying this derivative shows \(f'(x) < 0\) for sufficiently large \(x\), indicating \( f(x) \) is decreasing eventually.

Step 4: Evaluate Limit of Terms

Evaluate \( \lim_{n \to \infty} \frac{n \ln n}{2^n} \). As \( n \to \infty \), the exponential function in the denominator grows much faster than the polynomial in the numerator. Thus, \( \lim_{n \to \infty} \frac{n \ln n}{2^n} = 0 \).

Step 5: Apply the Alternating Series Test

Since \( \frac{n \ln n}{2^n} \) is decreasing for sufficiently large \(n\) and the limit of this sequence is zero, the conditions of the Alternating Series Test are satisfied. Therefore, the series converges.

Key Concepts

Series Convergence

In mathematics, when we talk about series convergence, we are essentially asking if the sum of a series has a finite value.
More intuitively, does the series "settle down" to a single number as we keep adding up its terms one by one?

• If it does, we say the series converges.
• If it doesn't, then it diverges and grows indefinitely or fluctuates without approaching a finite limit.

To determine whether a series converges, various tests can be used. One popular test for alternating series, like the one in the original exercise, is the Alternating Series Test. Checking these conditions helps us decide whether the infinite sum of the series approaches a fixed number as more terms are summed.

Alternating Series

An alternating series is a series whose terms alternate between positive and negative.
In an equation, we might see this alternation shown by a term like \((-1)^n\) or \((-2)^n\), where the base number induces the switch in signs.

• Alternating series can create special patterns that are interesting to analyze because their sum might converge when non-alternating series wouldn’t.
• Each subsequent term tries to "cancel out" a little more of the series, leading to the possibility of convergence even if the individual terms don't tend to zero quickly.

In our exercise, the series \(\sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}\) becomes an alternating series because of the \((-2)^n\) term.

Limiting Behavior

To decide if a series converges, particularly in the context of alternating series, we often examine the limiting behavior of its individual terms.
That is, we look at what happens to the terms as the index \(n\) gets very large.

• The key is for these terms to get progressively smaller, at least in terms of absolute value.
• If \(\lim_{n \to \infty} a_n = 0\), meaning the terms approach zero, it creates a scenario where adding more of these terms doesn't substantially change the total sum.

This limiting behavior is critical for the Alternating Series Test because it confirms that the series won't "explode" to an infinite sum but will stabilize.

Quotient Rule

The quotient rule is an essential calculus tool for finding the derivative of a function that is the ratio of two differentiable functions.
The quotient rule can sometimes help us check if a function is increasing or decreasing, which we need to do for testing convergence.

• If a function that's representative of a series term is decreasing, it fits one of the criteria for certain convergence tests like the Alternating Series Test.
• The rule is: if you have a function \(f(x) = \frac{g(x)}{h(x)}\), then its derivative is \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\).

In our exercise, we used the quotient rule on \(f(x) = \frac{x \ln x}{2^x}\) to determine that \(f(x)\) is decreasing for large \(x\), which helps prove convergence via the Alternating Series Test.