Problem 48 - Essential Calculus

Question

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(3 / 2)^{n}}{n^{2}} $$

Step-by-Step Solution

Answer

By applying the Ratio Test, it is determined that the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}(3 / 2)^{n}}{n^{2}}\) converges.

Step 1: Identify the series terms

First, identify the nth term of the series. In this case, it is given by: \[A_n = \frac{(-1)^{n-1}(3 / 2)^n}{n^{2}}\].

Step 2: Find the ratio of consecutive terms

The ratio of the (n+1)th term to the nth term: \[R = \left|\frac{A_{n+1}}{A_{n}}\right|\]. Replace \(A_{n+1}\) with the series formula replacing n with n+1 and \(A_{n}\) with the original series formula. The ratio \(R\) simplifies to: \[R = \left|\frac{3(n^{2})}{2(n+1)^{2}}\right|\].

Step 3: Determine the limit of the ratio as n approaches infinity

Calculate \[\lim_{n \to \infty}R\]. This simplifies to \[0\]. Since this limit is less than 1, the series converges.

Key Concepts

Convergence and Divergence of Series

Understanding whether a series converges or diverges is fundamental to grasping the behavior of infinite series. Convergence means that the sum of the series approaches a finite number as more terms are added. Conversely, divergence indicates that the sum increases without bound or behaves unpredictably as terms are added. One effective method for testing a series for convergence is the Ratio Test.

Using the Ratio Test, we compare the limit of the absolute ratios of successive terms in the series. If this limit is less than 1, the series converges absolutely; if it is greater than 1, the series diverges; and if the limit equals 1, the test is inconclusive. This is a powerful tool because it doesn’t require us to find the sum of the series to determine its nature, which is often more complex or even impossible to find.

Infinite Series

An infinite series is the sum of infinitely many terms that follow a particular sequence. It is written in the form \( \sum_{n=1}^{\infty} a_n \) where \(a_n\) represents the nth term of the series. These series can either converge to a specific value or diverge.

However, the term 'infinite' can be deceptive—while there are infinitely many terms, it doesn't mean the sum is infinite. For example, the series in our exercise shows that despite having infinite terms, it may still have a finite sum, as indicated by the convergence result from the Ratio Test. This is one of the intriguing aspects of infinite series that captivates mathematicians. They stand in contrast to finite series, where the sum is calculated over a specific number of terms.

Limit of a Sequence

The limit of a sequence is the value that the terms of a sequence approach as the index goes to infinity. Formally, if the limit of sequence \(a_n\) as \(n \) approaches infinity is \(L\), we denote it as \( \lim_{n \to \infty} a_n = L\). This concept is critical when analyzing series for convergence.

In the context of the Ratio Test, we looked at the limit of the ratio of consecutive terms to determine the series' behavior. If the terms of a sequence get infinitely close to a particular number, the sequence has a limit, which can play a crucial role in determining the sum of an infinite series. Understanding limits is essential in calculus because they form the foundation of many other concepts, including derivatives and integrals, which measure change and area, respectively.