Question

Explain how the Root Test works.

Short answer

Question: Explain how the Root Test works and how it is used to determine convergence or divergence of a series. Answer: The Root Test is a method used in calculus to determine the convergence or divergence of a series, specifically useful when the series has terms with powers or factorials. A series converges if the sum of its terms approaches a finite value as the number of terms increases, while it diverges if the sum does not approach a finite value. To use the Root Test, follow these steps: 1. Identify the series as \(\sum_{n=1}^{\infty} a_{n}\). 2. Calculate the limit of the nth root of the absolute value of the terms of the series: \(\lim_{n \to \infty} \sqrt[n]{|a_n|}\). 3. Analyze the result of the limit. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the Root Test is inconclusive, and another method must be used to determine the convergence or divergence of the series.

Step-by-step solution

Introducing the Root Test

The Root Test is a useful method for determining the convergence or divergence of a series. It is particularly beneficial when the series has terms with powers or factorials, which can make using other tests, such as the Ratio Test, difficult.

Convergence and Divergence

Before discussing the Root Test, it's important to understand the concepts of convergence and divergence. A series converges if the sum of its terms approaches a finite value as the number of terms increases. In contrast, a series diverges if the sum of its terms does not approach a finite value.

Root Test Statement

Suppose we have a series, \( \sum_{n=1}^{\infty} a_{n}\), with non-negative terms. The Root Test states that the series converges if: $$\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$$ The series diverges if: $$\lim_{n \to \infty} \sqrt[n]{|a_n|} > 1$$ If the limit is equal to 1, the Root Test is inconclusive, and another test must be used to determine the convergence or divergence of the series.

Identify the series

First, identify the series that you want to analyze, which can be written in the form \(\sum_{n=1}^{\infty} a_{n}\). This will serve as the basis for the Root Test.

Apply the Root Test

Next, calculate the limit of the nth root of the absolute value of the terms of the series: $$\lim_{n \to \infty} \sqrt[n]{|a_n|}$$

Determine convergence or divergence

Analyze the result of the limit calculated in Step 2. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the Root Test is inconclusive, and another method must be used to determine the convergence or divergence of the series.

Example: Application of the Root Test

Consider the series: \(\sum_{n=1}^{\infty} \frac{n^2}{3^n}\). 1. Identify the series: \(a_n = \frac{n^2}{3^n}\) 2. Apply the Root Test: Calculate \(\lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \sqrt[n]{\frac{n^2}{3^n}}\) 3. Simplifying the limit: \(\lim_{n \to \infty} \frac{\sqrt[n]{n^2}}{\sqrt[n]{3^n}} = \lim_{n \to \infty} \frac{\sqrt[n]{n^2}}{3}\) 4. Determine convergence or divergence: Since the limit of the denominator is fixed (3) and the limit \(\lim_{n \to \infty} \sqrt[n]{n^2}\) equals 1, the overall limit is less than 1. Hence, the series converges. By following these steps, you can use the Root Test to determine the convergence or divergence of a series.