Problem 42 - Essential Calculus Early Transcendentals

Question

\(41-42=\) Let \(\left\\{b_{n}\right\\}\) be a sequence of positive numbers that con- verges to \(\frac{1}{2} .\) Determine whether the given series is absolutely convergent. $$\sum_{n=1}^{\infty} \frac{(-1)^{n} n !}{n^{n} b_{1} b_{2} b_{3} \cdots b_{n}}$$

Step-by-Step Solution

Answer

The series is not absolutely convergent.

Step 1: Understand Absolute Convergence

For a series \(\sum a_n\) to be absolutely convergent, the series \(\sum |a_n|\) must converge. We must first examine the absolute value of the general term of the given series.

Step 2: Write the General Term

The general term of the given series is \(a_n = \frac{(-1)^{n} n!}{n^{n} b_1 b_2 \cdots b_n}\). Consequently, the absolute value of the general term is \(|a_n| = \frac{n!}{n^{n} b_1 b_2 \cdots b_n}\).

Step 3: Analyze Convergence of the Positive Series

To determine absolute convergence, we need to check whether the positive series \(\sum_{n=1}^{\infty} \frac{n!}{n^{n} b_1 b_2 \cdots b_n}\) converges. We know that \(b_n \to \frac{1}{2}\), meaning for large \(n\), each \(b_n\approx\frac{1}{2}\).

Step 4: Estimate the Terms

Since \(b_n \approx \frac{1}{2}\) for large \(n\), we approximate the product \(b_1b_2\cdots b_n \approx \left(\frac{1}{2}\right)^n\). Thus, the terms \(|a_n| \approx \frac{n!}{n^n \cdot (\frac{1}{2})^n} = \frac{n! \cdot 2^n}{n^n}.\)

Step 5: Apply Ratio Test for Convergence

The ratio test is useful here. We explore\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{(n+1)!}{(n+1)^{n+1} \cdot (\frac{1}{2})^{n+1}} \cdot \frac{n^{n} \cdot (\frac{1}{2})^n}{n!}\]This simplifies to a limit of \(\lim_{n \to \infty} \frac{(n+1) 2}{n+1} = 2 > 1\), indicating divergence.

Step 6: Conclusion

Since the ratio test yields a result greater than 1, the series \(\sum_{n=1}^{\infty} \frac{n!}{n^{n} b_1 b_2 \cdots b_n}\) diverges. Therefore, the original series is not absolutely convergent.

Key Concepts

Ratio Test for Convergence

Understanding when a series converges can be tricky, but the ratio test is a valuable tool in these investigations. The ratio test examines each term in the series in relation to the next term. By finding the limit of the ratio of consecutive terms as the sequence progresses to infinity, we can draw important conclusions. If this ratio is less than 1, the series converges absolutely. If it's greater than 1, the series diverges. When the ratio equals 1, the test is inconclusive.

In our exercise, we apply this test to the positive series formed by the non-negative absolute values of our original series terms. The formula applied was:\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]In this particular series, after simplifying, the ratio resulted in a limit of 2, which is greater than 1. Therefore, the test shows that the series diverges and does not converge absolutely.

Sequence of Positive Numbers

Sequences form the building blocks for understanding series. A sequence is simply a list of numbers in a specific order. When talking about a sequence of positive numbers, each term in the sequence is greater than 0.

In our original exercise, the sequence \({b_n}\) consists of positive numbers converging to \(\frac{1}{2}\). This means as \(n\) grows larger, the terms of this sequence get closer and closer to \(\frac{1}{2}\). Sequences approaching a specific limit help simplify terms when studying series, especially when determining convergence by approximation.

Factorial and Exponential Terms in Series

Factorials and exponentials often appear in mathematical series, adding a layer of complexity that can challenge understanding. A factorial, denoted \(n!\), is the product of all positive integers up to \(n\), growing extremely fast as \(n\) increases. Exponential terms such as \(n^n\) also grow rapidly, often faster than factorials, especially when large bases are repeatedly multiplied.

In the exercise's given series, the term specific structure combines factorials \(n!\) and exponentials \(n^n\). Understanding how each grows is crucial to simplifying and comparing terms within the series. This is especially important when using the ratio test or other methods of analysis, as the balance between these functions can quickly determine the series' behavior regarding convergence or divergence.