Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{e^{n}} $$

Step-by-Step Solution

Answer

The series converges absolutely by the ratio test.

Step 1: Identify the Series Type

The series given is \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{e^{n}} \). This is an alternating series, where each term has an alternating sign due to \((-1)^{n+1}\). Our goal is to check if this series converges absolutely by considering the absolute value \( \sum_{n=1}^{\infty} \left| (-1)^{n+1} \frac{n^{2}}{e^{n}} \right| = \sum_{n=1}^{\infty} \frac{n^{2}}{e^{n}} \).

Step 2: Apply the Ratio Test

To determine if \( \sum_{n=1}^{\infty} \frac{n^{2}}{e^{n}} \) converges, we apply the ratio test. Define \( a_n = \frac{n^{2}}{e^{n}} \). The ratio test involves calculating the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

Step 3: Compute the Ratio

Calculate \( \frac{a_{n+1}}{a_n} = \frac{(n+1)^2/e^{n+1}}{n^2/e^n} = \frac{(n+1)^2}{e} \times \frac{e^n}{n^2} = \frac{(n+1)^2}{en^2} \).

Step 4: Evaluate the Limit

Calculate the limit \( L = \lim_{n \to \infty} \frac{(n+1)^2}{en^2} = \lim_{n \to \infty} \frac{n^2 + 2n + 1}{en^2} = \lim_{n \to \infty} \frac{1 + \frac{2}{n} + \frac{1}{n^2}}{e} \). As \( n \to \infty \), the terms \( \frac{2}{n} \) and \( \frac{1}{n^2} \) go to zero. Thus, \( L = \frac{1}{e} < 1 \).

Step 5: Conclude the Convergence

Since the limit \( L < 1 \), by the ratio test, the series \( \sum_{n=1}^{\infty} \frac{n^{2}}{e^{n}} \) converges. This means that the original alternating series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{e^{n}} \) converges absolutely because the series of their absolute values converges.

Key Concepts

Alternating Series

An alternating series is a series where the terms alternate in sign. In our given series, denoted as \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{e^{n}} \), the factor \((-1)^{n+1}\) is responsible for this alternation. Each term flips between positive and negative as \(n\) increases.

Alternating series can converge absolutely or conditionally. For absolute convergence, the absolute value series must converge: \( \sum_{n=1}^{\infty} \left|(-1)^{n+1} \frac{n^{2}}{e^{n}}\right| = \sum_{n=1}^{\infty} \frac{n^{2}}{e^{n}} \).

• Absolute Convergence: The series formed with absolute values must converge.
• Conditional Convergence: The original series converges but not the absolute value series.

Recognizing the nature of the series is crucial to applying tests like the ratio test or other convergence criteria.

Ratio Test

The ratio test is a common method to determine the convergence of a series. It considers the limit of the absolute ratio of consecutive terms. If this limit, \(L\), is less than 1, the series converges absolutely.

For our series \(\sum_{n=1}^{\infty} \frac{n^{2}}{e^{n}} \), define \(a_n = \frac{n^{2}}{e^{n}}\). We calculate the ratio \( \frac{a_{n+1}}{a_n} = \frac{(n+1)^2}{en^2} \) and evaluate:

• Apply the Ratio Test: Calculate \( L = \lim_{n \to \infty} \frac{(n+1)^2}{en^2} \).
• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

The ratio test is especially useful when dealing with exponential and factorial terms, as it simplifies comparison of the growth rates of the terms.

Limit Evaluation

Limit evaluation involves calculating the limit of a function as \(n\) approaches infinity. In this context, it's about understanding how the series behaves in the long run.

For our series, the limit \( L \) is based on \(\lim_{n \to \infty} \frac{(n+1)^2}{en^2} = \lim_{n \to \infty} \frac{1 + \frac{2}{n} + \frac{1}{n^2}}{e} \). As \(n\) becomes very large, the fractional terms \(\frac{2}{n}\) and \(\frac{1}{n^2}\) become negligible.

• The limit simplifies to \( \frac{1}{e} \).
• Since \( \frac{1}{e} < 1 \), it confirms the absolute convergence.

Evaluating such limits can give us a clear picture of the behavior of the terms as \(n\) increases indefinitely.

Series Convergence

Series convergence is the idea that the sum of infinite terms can approach a finite value. Different tests, such as the ratio test, help us analyze whether a series converges.

In our given series, we explored convergence through the steps of identifying it as an alternating series and applying the ratio test. Absolute convergence indicates this series not only converges in its alternating form but also when considering the sum of absolute values.

• Ratio Test Result: The ratio test applied to the series of absolute values revealed \( L = \frac{1}{e} \).
• Conclusion: \( L < 1 \) confirmed the absolute convergence.

Convergence reveals the stability and finite sum nature of infinite series, providing a foundational tool in analysis and calculus.