Question

Explain how the Ratio Test works.

Short answer

Question: Explain the purpose and the process of the Ratio Test for determining the convergence of an infinite series. Answer: The Ratio Test is a convergence test for infinite series, used to determine whether a given series converges or diverges. The process involves calculating the limit L of the absolute value of the ratio between consecutive terms of the series and then applying specific rules based on the value of L. If L < 1, the series converges absolutely, if L > 1, the series diverges, and if L = 1, the test is inconclusive. The Ratio Test is particularly useful when dealing with series involving exponentials or factorials.

Step-by-step solution

Introduction

The Ratio Test is a useful convergence test for infinite series. Given a series ∑ a_n, the Ratio Test is used to determine whether the series converges or diverges.

Define the Limit

Start by calculating the limit L of the absolute value of the ratio between consecutive terms of the series: L = lim (n → ∞) [|a_(n+1)| / |a_n|]

Apply the Ratio Test

Based on the value of L calculated in the previous step, apply the following rules to determine if the series converges or diverges: - If L < 1, the series converges absolutely. - If L > 1, the series diverges. - If L = 1, the Ratio Test is inconclusive and we cannot make a determination about convergence or divergence.

Example

Let's illustrate the Ratio Test with an example. Consider the series: ∑_(n=1)^(∞) ((-1)^n n) / (3^n)

Calculate the Limit

L = lim (n → ∞) [|a_(n+1)| / |a_n|] L = lim (n → ∞) [|(((-1)^(n+1) (n+1)) / (3^(n+1)))| / |(((-1)^n n) / (3^n))|]

Simplify the Expression

By simplifying the above expression, we get: L = lim (n → ∞) [|(n+1) / (3^(n+1))| / |(n / (3^n))| ] L = lim (n → ∞) [|(n+1) / (3^(n+1))| * |(3^n) / n| ] L = lim (n → ∞) [|n+1| / |3n|] = lim (n → ∞) [(n+1) / (3n)]

Calculate the Limit

L = lim (n → ∞) [(n+1) / (3n)] = 1/3

Apply the Ratio Test

Since L < 1 (1/3 < 1), the series converges absolutely. In conclusion, the Ratio Test is a powerful tool for testing the convergence of infinite series, particularly when dealing with series involving exponentials or factorials. By following the steps outlined above and applying the Ratio Test, one can quickly determine if a series converges or diverges.