Problem 27 - Thomas Calculus

Question

Which of the series in Exercises \(15 - 48\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } n ^ { 2 } ( 2 / 3 ) ^ { n } $$

Step-by-Step Solution

Answer

The series converges absolutely by the ratio test.

Step 1: Analyze Absolute Convergence

To determine if the series \( \sum_{n=1}^{\infty} (-1)^n n^2 (2/3)^n \) converges absolutely, we first evaluate the series without the alternating sign, which is \( \sum_{n=1}^{\infty} n^2 \left( \frac{2}{3} \right)^n \). For absolute convergence, this series itself must converge.

Step 2: Apply the Ratio Test

To test convergence of \( \sum_{n=1}^{\infty} n^2 \left( \frac{2}{3} \right)^n \), we use the ratio test. Consider the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \) where \( a_n = n^2 \left( \frac{2}{3} \right)^n \).\[\left| \frac{(n+1)^2 \left( \frac{2}{3} \right)^{n+1}}{n^2 \left( \frac{2}{3} \right)^n} \right| = \left( \frac{n+1}{n} \right)^2 \cdot \frac{2}{3}\]This simplifies to \( \left( 1 + \frac{1}{n} \right)^2 \cdot \frac{2}{3} \). As \( n \to \infty \), this approaches \( \frac{2}{3} \).

Step 3: Conclude Ratio Test Analysis

Since \( \frac{2}{3} < 1 \), the ratio test indicates that the series \( \sum_{n=1}^{\infty} n^2 \left( \frac{2}{3} \right)^n \) converges. Therefore, the original series \( \sum_{n=1}^{\infty} (-1)^n n^2 (2/3)^n \) converges absolutely.

Key Concepts

Absolute Convergence

Absolute convergence means that a series converges even when we take the absolute value of its terms. In our case, we have an alternating series \( \sum_{n=1}^{\infty} (-1)^n n^2 (2/3)^n \). To test for absolute convergence, we remove the alternating factor \((-1)^n\), and consider \( \sum_{n=1}^{\infty} n^2 (2/3)^n \). If this new series without the sign converges, then our original series is deemed to converge absolutely.

This process ensures that the series remains convergent regardless of the signs of its terms, making it easier to analyze without worrying about cancellation of terms.

Ratio Test

The Ratio Test is a popular method for determining the convergence of series. It involves computing the limit of the ratio of consecutive terms. For the series \( \sum_{n=1}^{\infty} n^2 (2/3)^n \), the test is as follows: consider the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \), where \( a_n = n^2 (2/3)^n \).

The formula becomes:

• \( \left| \frac{(n+1)^2 (2/3)^{n+1}}{n^2 (2/3)^n} \right| = \left( \frac{n+1}{n} \right)^2 \cdot \frac{2}{3} \)

Simplifying gives:

• \( \left( 1 + \frac{1}{n} \right)^2 \cdot \frac{2}{3} \)

As \( n \to \infty \), this expression approaches \( \frac{2}{3} \), which is less than 1. By the Ratio Test, the series converges due to the limit being less than 1.

Alternating Series

Alternating series are those where the terms alternate in sign, such as \( (-1)^n \) in our original series. These series can still converge even if they don't converge absolutely. The Alternating Series Test gives conditions for their convergence, focusing on:

• The absolute value of the terms must be decreasing, meaning \( |a_{n+1}| < |a_n| \).
• The limit of the absolute value of the terms as \( n \to \infty \) must be zero, i.e., \( \lim_{n \to \infty} |a_n| = 0 \).

Our original series meets these conditions, and with absolute convergence already proven, it ensures overall convergence too.

Convergence and Divergence

Convergence and divergence are fundamental concepts when analyzing series. A series converges if the sum approaches a specific number, while it diverges if the sum does not settle on any specific value. For series like the one given, different tests like the Ratio Test and Absolute Convergence check can provide insight into whether the series converges.

When a series is positively convergent, it opens up easier paths to compute values and analyze the behavior of functions related to these series. Divergence denotes instability in terms of long-term behavior, often leading to sums that grow unbounded or fluctuate indefinitely.

Infinite Series Analysis

Infinite series analysis involves comprehensively understanding series that extend indefinitely. It requires a toolkit of tests and concepts like those mentioned to check convergence, such as the Ratio Test and the Alternating Series Test. They help us understand the behavior of infinte sums, both practically and theoretically.

Analyzing infinite series has applications across mathematics and physics, such as in Taylor series, Fourier series, and quantum mechanics. Grasping these concepts aids in making predictions and solving complex problems by reliably understanding the underlying infinite behavior. This reflects the importance of assessing series accurately, ensuring the sums meet our expectations, be it in theoretical computations or real-world applications.