Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 13–30, classify each series as absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{4}}{2^{n}} $$

Step-by-Step Solution

Answer

The series is absolutely convergent.

Step 1: Understand the Series

The given series is \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{4}}{2^{n}} \). It is an alternating series because it includes the factor \((-1)^{n+1}\). To determine convergence, we need to check both absolute convergence and conditional convergence.

Step 2: Test for Absolute Convergence

To check for absolute convergence, consider the series \( \sum_{n=1}^{\infty} \left|(-1)^{n+1} \frac{n^{4}}{2^{n}}\right| \), which simplifies to \( \sum_{n=1}^{\infty} \frac{n^{4}}{2^{n}} \). We will use the Ratio Test for convergence.

Step 3: Apply the Ratio Test for Absolute Convergence

Compute the ratio \( \frac{a_{n+1}}{a_n} \) where \( a_n = \frac{n^4}{2^n} \):\[ \lim_{n \to \infty} \frac{(n+1)^4}{2^{n+1}} \cdot \frac{2^n}{n^4} = \lim_{n \to \infty} \frac{(n+1)^4}{2 n^4} = \frac{1}{2} \times \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^4 = \frac{1}{2} \times 1 = \frac{1}{2}. \]Since \( \frac{1}{2} < 1 \), the series \( \sum_{n=1}^{\infty} \frac{n^4}{2^n} \) is absolutely convergent.

Step 4: Conclusion on Convergence Type

Since the series \( \sum_{n=1}^{\infty} \frac{n^4}{2^n} \) is absolutely convergent, it follows that the original series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^4}{2^n} \) is also absolutely convergent. Thus, there is no need to check for conditional convergence.

Key Concepts

Alternating Series

An alternating series is a series where the terms alternate in sign. This alternation is typically due to a factor like \((-1)^{n+1}\). Such series can often be recognized by this alternating sign factor. Alternating series play an important role in understanding series convergence because they can converge even if their corresponding series of absolute values does not.

For alternating series, a useful test is the **Alternating Series Test**. A series \(\sum_{n=1}^{\infty} (-1)^{n} b_n\) converges if

• The absolute value of the terms, \(b_n\), decreases steadily. That is, \(b_{n+1} \leq b_n\) for all \(n\).
• The limit of \(b_n\) as \(n\) goes to infinity is zero: \(\lim_{n \to \infty} b_n = 0\).

Using this test, one can determine that the alternating series converges if both conditions are fulfilled. However, do note in our problem, we used the **Ratio Test** for checking absolute convergence rather than directly using the Alternating Series Test for convergence conditions.

Absolute Convergence

The concept of absolute convergence is crucial because if a series converges absolutely, then it converges under all conditions. To test for absolute convergence, you take the absolute value of each term in the series and determine whether this new series converges.

• If the series of absolute values \(\sum |a_n|\) converges, the original series \(\sum a_n\) is said to be absolutely convergent.
• An absolutely convergent series converges regardless of the order of its terms, which isn't necessarily true for conditionally convergent series.

In our exercise, the given series was \(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^4}{2^n}\). To assess absolute convergence, we looked at \(\sum_{n=1}^{\infty} \frac{n^4}{2^n}\) and found it to converge. Hence, the original series is absolutely convergent. This simplifies our solution by confirming convergence without additional checks for conditional convergence.

Ratio Test

The Ratio Test is one of the handy tools to determine the convergence of a series. It involves taking the limit of the absolute value of the ratio of consecutive terms of the series as \(n\) approaches infinity.

• We express this as \(|\frac{a_{n+1}}{a_n}|\).
• If this limit is less than 1, the series converges absolutely.
• If greater than 1, the series diverges.
• If exactly 1, the test is inconclusive.

In our example, we applied the Ratio Test to the series \(\sum_{n=1}^{\infty} \frac{n^4}{2^n}\). By computing \(\lim_{n \to \infty} \frac{(n+1)^4}{2^{n+1}} \cdot \frac{2^n}{n^4}\), we simplified to get a limit of 1/2, which is less than 1. Consequently, the series is absolutely convergent, supporting the conclusion about the behavior of the original series under scrutiny. The Ratio Test, thus, confirmed our series converges absolutely, allowing us to confidently state the original series' convergence outcome.