Problem 42 - Thomas Calculus

Question

In Exercises \(17-46,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3} 2^{n}} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify the Series Components

The given series is \( \sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3} 2^{n}} \). This is an alternating series since the term \((-3)^n\) introduces a change in sign as \(n\) progresses. Let's examine the possibility of applying the Alternating Series Test.

Step 2: Alternating Series Test Conditions

The Alternating Series Test states that the series \( \sum (-1)^n a_n \) converges if \( a_n > 0 \), \( a_n \) is decreasing, and \( \lim_{n \to \infty} a_n = 0 \). In this case \( a_n = \frac{3^n}{n^3 2^n} = \left(\frac{3}{2}\right)^n \frac{1}{n^3} \). Now, we need to check these conditions.

Step 3: Check if Each Term is Positive

For \( a_n \) (which is \( \frac{3^n}{n^3 2^n} \)), each term must be positive for the alternating series test. Since both the numerator and denominator are positive as \( n > 0 \), \( a_n > 0 \) for all \( n \).

Step 4: Check if the Sequence is Decreasing

Examine if \( a_n = \frac{3^n}{n^3 2^n} \) is decreasing by analyzing \( \frac{a_{n+1}}{a_n} \): \[ \frac{3^{n+1}/((n+1)^3 2^{n+1})}{3^n/(n^3 2^n)} = \left(\frac{3}{2}\right) \left(\frac{n^3}{(n+1)^3}\right). \] As \( n \) increases, \( \frac{n^3}{(n+1)^3} \to 1 \). Therefore, each term decreases only if \( \frac{3}{2} \frac{n^3}{(n+1)^3} < 1 \), which doesn't hold for all \( n \). So \( a_n \) is not guaranteed to decrease monotonically.

Step 5: Limit of the Terms

Evaluate \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(\frac{3}{2}\right)^n \frac{1}{n^3} \). Since \( \left(\frac{3}{2}\right)^n \to \infty \) faster than \( n^3 \), \( a_n \to \infty \), not zero, which violates a condition of the Alternating Series Test.

Step 6: Conclusion

As the alternating series test fails due to a nonzero term limit, and the ratio \( \left(\frac{3}{2}\right) > 1 \), which dominates, the series \( \sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3} 2^{n}} \) diverges.

Key Concepts

Alternating Series Test

The Alternating Series Test is a valuable tool when analyzing whether certain series converge. It applies to series with terms that alternate in sign, meaning that the signs of the terms switch back and forth as the series progresses. Such series are known as alternating series.

The test states that an alternating series \( \sum (-1)^n a_n \) converges if the following conditions are satisfied:

• Each term \( a_n \) is positive.
• The sequence of terms \( a_n \) is decreasing.
• The limit of the terms as \( n \) approaches infinity is zero: \( \lim_{n \to \infty} a_n = 0 \).

In the given problem, we are looking at a series where the sequence \( a_n = \frac{3^n}{n^3 2^n} \). For convergence via the alternating series test, while the terms are positive, they must also decrease uniformly and ultimately tend to zero, conditions not met here.

Limit Comparison Test

The Limit Comparison Test is a versatile technique used to compare two series to determine their convergence properties. It works by analyzing the ratio of the terms of one series to another. We apply this test typically to series that are positive term series, meaning each term is greater than zero.

To implement the Limit Comparison Test:

• Select a series \( \sum b_n \) that is known to converge or diverge.
• Compute the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \), where \( a_n \) are the terms of the given series.
• If this limit is a positive finite number, then both series \( \sum a_n \) and \( \sum b_n \) converge or diverge together.

This test helps in cases where an unknown series \( \sum a_n \) resembles a known series, facilitating conclusions about their convergence behavior.

Divergence

In mathematics, divergence refers to a series that does not converge to a finite value. If a series diverges, it means that as you keep adding more and more terms, the series either grows indefinitely or fails to settle at a specific number.

One straightforward test for divergence is checking the limit of the sequence terms: if \( \lim_{n \to \infty} a_n eq 0 \), then the series \( \sum a_n \) definitely diverges. In our exercise, the fact that \( \lim_{n \to \infty} \left( \frac{3}{2} \right)^n \frac{1}{n^3} = \infty \) leads directly to the conclusion that the series diverges.

Understanding divergence is crucial as it tells us that no matter how many terms we add together, the series will never reach a specific total sum.

Infinite Series

An infinite series is the sum of an infinite sequence of numbers. Infinite series can converge to a finite value or diverge, depending on the behavior of their terms over the long run.

They are represented as \( \sum_{n=1}^{\infty} a_n \), where each \( a_n \) is a term in the sequence. The challenge is determining whether this sum is finite or not.

The methodology for handling infinite series involves various tests, such as the Alternating Series Test or the Limit Comparison Test, to assess the collective behavior of the terms. In the exercise provided, the infinite series examined was \( \sum_{n=1}^{\infty} \frac{(-3)^n}{n^3 2^n} \), which ultimately diverged. Infinite series are foundational in advanced mathematics for modeling and solving complex problems in calculus and beyond.

Sequence Behavior

The behavior of a sequence is fundamental to understanding the series it sums to. A sequence is an ordered list of numbers, and examining how these numbers behave as the sequence progresses is key in determining series convergence.

There are several aspects of sequence behavior in focus:

• Positivity: Ensuring each term is positive, particularly in tests like the Alternating Series Test.
• Monotonicity: Checking if the sequence terms increase or decrease consistently, which impacts certain convergence tests.
• Limits: Determining what happens to terms as \( n \to \infty \) helps us understand if sequence terms approach zero, significantly influencing the conclusion about a series' convergence or divergence.

In our specific problem, the sequence \( a_n = \frac{3^n}{n^3 2^n} \) did not decrease monotonically and did not tend towards zero, contributing to the overall divergent nature of the series.