Problem 16 - Thomas Calculus

Question

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n^{1+n}}$$

Step-by-Step Solution

Answer

The series converges absolutely.

Step 1: Identify the Series

Given the series \(\sum_{n=2}^{\infty} \frac{(-1)^n}{n^{1+n}}\), we first identify the general term \(a_n = \frac{(-1)^n}{n^{1+n}}\). We will apply the Root Test, which is useful for series with terms raised to the power of \(n\).

Step 2: Apply the Root Test Formula

The Root Test involves taking the \(n\)-th root of the absolute value of \(a_n\). This means we compute \(\lim_{n \to \infty} \sqrt[n]{|a_n|}\). Here, \(|a_n| = \frac{1}{n^{1+n}}\), so we consider \(\lim_{n \to \infty} \sqrt[n]{\frac{1}{n^{1+n}}}\).

Step 3: Simplify the Expression

Since \(|a_n| = \frac{1}{n^{1+n}} = \frac{1}{n \cdot n^n} = \frac{1}{n^{n+1}} = n^{-(n+1)}\), the root expression becomes \((\frac{1}{n^{n+1}})^{1/n} = \frac{1}{n^{(n+1)/n}} = \frac{1}{n^{1 + 1/n}}\).

Step 4: Evaluate the Limit

Now, evaluate \(\lim_{n \to \infty} \frac{1}{n^{1 + 1/n}}\). As \(n \to \infty\), the term \(1/n\) tends to zero. Thus, this expression simplifies to \(\lim_{n \to \infty} \frac{1}{n}\), which equals \(0\).

Step 5: Conclusion with the Root Test

Since \(\lim_{n \to \infty} \sqrt[n]{|a_n|} = 0 < 1\), by the Root Test, the series \(\sum_{n=2}^{\infty} \frac{(-1)^n}{n^{1+n}}\) converges absolutely.

Key Concepts

Convergence and Divergence of Series

Understanding when an infinite series converges or diverges is an essential part of mathematical analysis. When a series converges, the sum of its terms approaches a finite value. If it diverges, the sum grows without bound or does not reach a finite limit.

The Root Test is a powerful tool for examining convergence of series, particularly where terms include powers of the index, like those of the form \(a_n^n\).

To use the Root Test:

• Calculate the \(n\)-th root of the absolute value of the general term, \(|a_n|\).
• Evaluate the limit \(\lim_{n \to \infty} \sqrt[n]{|a_n|}\).

If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. If equal to 1, the test is inconclusive.
In the example provided, the limit is calculated as 0, which indicates convergence.

Absolute Convergence

A critical concept in series analysis is absolute convergence. A series \(\sum a_n\) converges absolutely if the series of absolute values \(\sum |a_n|\) also converges.

If a series converges absolutely, it is guaranteed to converge in the real number sense, which means the arrangement of terms doesn't affect the limit. This is a stronger condition compared to regular convergence.

For the given example, applying the Root Test showed the limit of \(\sqrt[n]{|a_n|}\) is zero. Since this is indeed less than one, the original series \(\sum_{n=2}^{\infty} \frac{(-1)^n}{n^{1+n}}\) converges absolutely.

Infinite Series Analysis

Infinite series analysis involves a detailed study of series to determine their behavior. This typically includes tests for convergence and divergence, identifying whether sums approach a finite limit.

The Root Test is particularly effective for series with exponential terms, where working with absolute values can simplify analysis. This requires a deep understanding of limits and power rules.
When performing an infinite series analysis, it’s helpful to:

• Recognize patterns or forms of series, such as geometric or p-series.
• Apply appropriate convergence tests like the Root Test or Ratio Test.
• Use absolute values to transform terms, making them easier to evaluate.

In the context of this analysis, knowing how to efficiently apply these strategies ensures an accurate understanding of whether and how an infinite series converges or diverges.