Problem 38 - Thomas Calculus

Question

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n !}{(-n)^{n}}$$

Step-by-Step Solution

Answer

The series converges by the Ratio Test.

Step 1: Understanding the Series

The given series is \( \sum_{n=1}^{\infty} \frac{n!}{(-n)^{n}} \). This is an infinite series where each term is of the form \( \frac{n!}{(-n)^{n}} \). Our task is to determine whether this series converges or diverges.

Step 2: Analyzing the Terms

Firstly, consider the sign of each term. Since \((-n)^{n}\) is negative whenever \(n\) is odd, this series alternates in sign. An alternating series might converge, so further investigation is needed to check whether the absolute values of the terms \( \left|\frac{n!}{n^n}\right| \) tend to zero.

Step 3: Apply the Ratio Test

To determine the nature of the series, apply the Ratio Test. Consider the ratio of consecutive terms:\[ a_n = \frac{n!}{(-n)^n} \text{ and } a_{n+1} = \frac{(n+1)!}{(-(n+1))^{n+1}} \]Calculate the ratio:\[ \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(n+1)!}{(-(n+1))^{n+1}} \cdot \frac{(-n)^n}{n!} \right| \]Simplify this to:\[ \frac{(n+1) \cdot n^n}{n^{n+1}} = \frac{n+1}{n+1} = \frac{1}{e} \rightarrow 0 \text{ as } n \rightarrow \infty \]

Step 4: Result from the Ratio Test

The Ratio Test indicates that if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \), the series converges. Since our limit of the ratio is \( \frac{1}{e} < 1 \), the series converges.

Key Concepts

Ratio Test

The Ratio Test is a powerful method to assess the convergence or divergence of an infinite series. Specifically, it evaluates the limit of the absolute value of the ratio between consecutive terms in a series to determine its behavior. To apply the Ratio Test:

• First, identify the series and express its general term, denoted as \( a_n \).
• Calculate the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \).
• Evaluate the limit of this ratio as \( n \) approaches infinity.

If the limit \( L \) of the ratio:

• is less than 1, the series converges.
• is greater than 1, the series diverges.
• is equal to 1, the test is inconclusive, meaning we need to use another test.

In our problem, we calculated the limit to be \( \frac{1}{e} \), which is less than 1. Hence, the series converges according to the Ratio Test.

Alternating Series

An alternating series is one where the terms alternate in sign. This characteristic can sometimes aid in the convergence of a series. Here are key points:

• The general form for an alternating series is \( \sum (-1)^n a_n \).
• Such series can start with positive or negative terms, but the key is the switch in sign.

To determine convergence of an alternating series, we use the Alternating Series Test, which states that the series \( \sum (-1)^n a_n \) converges if:

• The absolute value of the terms \( a_n \) decreases monotonically (each term is smaller than the previous term).
• The limit of \( a_n \) as \( n \to \infty \) is zero.

In our exercise, the series has alternating signs due to the presence of \((-n)^n\). This suggests we should explore both sign and absolute value in relation to convergence.

Infinite Series

An infinite series is simply the sum of infinitely many terms. Usually, it takes the form \( \sum_{n=1}^{\infty} a_n \). Key aspects to explore in infinite series include:

• The nature of the general term \( a_n \).
• Convergence: checking if the sum approaches a finite value as more terms are added.
• Divergence: where the sum becomes infinitely large, undefined, or oscillates without settling.

Various tests other than the Ratio Test can assist in assessing convergence of an infinite series, such as the Comparison Test, Integral Test, and Root Test.Understanding the behavior of the series is crucial; often, transformations or simplifications can reveal whether the series converges. In our example, despite being infinite, we applied the Ratio Test to analyze convergence conclusively.