Question

Explain how the Ratio Test works.

Short answer

Question: Explain the Ratio Test and how it's used to determine the convergence or divergence of an infinite series. Answer: The Ratio Test is used to analyze the convergence or divergence of an infinite series with positive terms. It involves calculating the limit \(L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}\) using the series' general term. Once the limit is found, we can determine the convergence or divergence based on the value of \(L\): if \(L < 1\), the series converges; if \(L > 1\), the series diverges; if \(L = 1\), the test is inconclusive, and another method should be used. The ratio test is not applicable to series with a mix of positive and negative terms.

Step-by-step solution

Definition of the Ratio Test

The Ratio Test states that given an infinite series \(\sum_{n=1}^\infty a_n\), with positive terms, we can determine its convergence or divergence by calculating the limit \(L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}\).

Conditions for Convergence and Divergence

Depending on the value of \(L\), we can determine whether the series converges or diverges: 1. If \(L < 1\), the series converges. 2. If \(L > 1\), the series diverges. 3. If \(L = 1\), the Ratio Test is inconclusive, and we need to use another method to determine convergence or divergence.

Apply the Ratio Test

To apply the Ratio Test to a given series, first write down the general term of the series \(a_n\), and then find the term \(a_{n+1}\). After that, form a ratio \(\frac{a_{n+1}}{a_n}\).

Simplify the Ratio

Simplify the ratio \(\frac{a_{n+1}}{a_n}\) as much as possible, and make sure it's in a form that allows you to easily calculate the limit as \(n\) approaches infinity.

Find the Limit

Calculate the limit \(L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}\) using algebraic techniques (L'Hopital's rule, factoring, or others).

Determine Convergence or Divergence

Based on the value of \(L\), determine if the given series converges or diverges using the conditions mentioned earlier: 1. If \(L < 1\), the series converges. 2. If \(L > 1\), the series diverges. 3. If \(L = 1\), the Ratio Test is inconclusive, and another method needs to be used to determine convergence or divergence. Note: It's important to keep in mind that the Ratio Test is only applicable to series with positive terms. For series with a mix of positive and negative terms, you may need to try another test like the Alternating Series Test or the Absolute Convergence Test.