Problem 26 - Thomas Calculus

Question

Which of the series in Exercises \(15 - 48\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } ( \sqrt [ n ] { 10 } ) $$

Step-by-Step Solution

Answer

The series diverges because the limit of the terms is not zero, failing the Alternating Series Test.

Step 1: Identify the Series Type and Apply Absolute Convergence Test

The given series is: \( \sum_{n = 1}^{\infty} (-1)^{n + 1} \sqrt[n]{10} \). This is an alternating series. First, we check for absolute convergence by considering the absolute value of the terms, \( \sum_{n = 1}^{\infty} \sqrt[n]{10} \). If this series converges, then our original series converges absolutely.

Step 2: Test for Absolute Convergence with Root Test

To apply the Root Test, compute \( \lim_{n \to \infty} \sqrt[n]{\sqrt[n]{10}} = \lim_{n \to \infty} 10^{1/n^2} \). This simplifies to \( 10^0 = 1 \). Since the limit is NOT less than 1, the series \( \sum_{n = 1}^{\infty} \sqrt[n]{10} \) does not converge, thus the original series does not converge absolutely.

Step 3: Check Conditional Convergence Using Alternating Series Test

Apply the Alternating Series Test to the original series, which has terms \((-1)^{n+1} \sqrt[n]{10}\). Verify the conditions: (1) \(\sqrt[n]{10} \to 1\), indeed the sequence is decreasing after a few initial terms and (2) \(\lim_{n\to\infty} \sqrt[n]{10} = 1\), which is not zero. Since the limit is not zero, the series diverges.

Key Concepts

Absolute Convergence

Absolute convergence is a crucial concept when dealing with series. It means that if you take the absolute value of each term in the series, and the new series (formed by these absolute values) converges, then the original series also converges. This type of convergence ensures stability in the behavior of the series.In the given exercise, we attempted to test for absolute convergence by evaluating the series \( \sum_{n = 1}^{\infty} \sqrt[n]{10} \).However, through the root test, we found that the limit of the series does not converge to something less than 1. Therefore, our series does not absolutely converge.Remember, absolute convergence implies the series is well-behaved, having none of the erratic swings or divergences that might occur otherwise. It's like ensuring the series behaves even when applying any operations without needing to worry about its stability.

Root Test

The Root Test, also known as the nth Root Test, is a helpful method for determining the convergence or divergence of series, especially when the series involves powers.To use the root test, consider a series \( \sum a_n \).We calculate \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \).- If the limit is less than 1, the series converges absolutely.- If the limit is greater than 1, the series diverges.- If the limit equals 1, the test is inconclusive.In our exercise, applying the root test yields a limit of 1 for the series \( \sum_{n=1}^{\infty} \sqrt[n]{10} \).Since the result is not less than 1, we cannot conclude that the series converges absolutely through this test.

Conditional Convergence

Conditional convergence occurs when a series converges, but it does not converge absolutely. It's like saying the series comes together under certain conditions, but if you consider just the magnitudes of terms, it fails to do so.In the context of our series, \[ \sum_{n = 1}^{\infty} (-1)^{n + 1} \sqrt[n]{10} \],to check for conditional convergence, we applied the Alternating Series Test.This test checks two main conditions:

• The absolute value of terms decreases monotonically (eventually)
• The limit of terms approaches zero as \(n\) heads to infinity

Unfortunately, the series fails in the second condition, as the limit does not approach zero. Therefore, although it's an alternating series, it does not converge even conditionally.

Divergence of Series

Divergence tells us that a series does not have a finite sum. This means no matter how many terms you add up, the series will not settle to a single value. It keeps growing indefinitely or does not stabilize.In the exercise, after applying tests for both absolute and conditional convergence, we found the series does not meet the necessary criteria for convergence.Therefore, it diverges.This result means that the series \( \sum_{n=1}^{\infty} (-1)^{n+1} \sqrt[n]{10} \)lacks a definitive total.Understanding divergence helps in recognizing when a series will not logically conclude to a real number, a crucial insight in advanced mathematical analysis.