Problem 98 - Essential Calculus

Question

Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. $$ 1+\frac{2}{3}+\frac{3}{3^{2}}+\frac{4}{3^{3}}+\frac{5}{3^{4}}+\frac{6}{3^{5}}+\cdots $$

Step-by-Step Solution

Answer

Using the Ratio Test, it is determined that the given series converges.

Step 1: Understanding the sequence terms

The terms in the series are given by \(a_n = \frac{n}{3^n}\). Thus, the next term in the series is \(a_{n+1} = \frac{n+1}{3^{n+1}}\).

Step 2: Setting up the Ratio Test

The Ratio Test compares the (n+1)th term of a sequence to the nth term. Write down the absolute ratio of the consecutive terms as \(\frac{a_{n+1}}{a_n}\). This corresponds to \(\frac{(n+1)/3^{n+1}}{n/3^n}\). Simplify this expression by cancelling out similar terms.

Step 3: Performing the Computation

Solving the ratio, you get \(\frac{n+1}{3n}\). As n approaches infinity, the fraction tends to \(1/3\).

Step 4: Interpreting the Result

According to the Ratio Test, if the limit of the ratio is less than 1, the series converges. Since \(1/3 < 1\), it can be concluded that this series converges.

Key Concepts

Ratio Test

The Ratio Test is a method used to determine the convergence or divergence of an infinite series. It specifically examines the absolute ratio of successive terms. In formal terms, for a given series \(\sum a_n\), the Ratio Test entails evaluating the limit \(L = \lim_{n \to \infty} |a_{n+1}/a_n|\).

Here's what the results mean:

• If \(L < 1\), the series converges.
• If \(L > 1\), the series diverges.
• If \(L = 1\), the test is inconclusive, and the convergence cannot be determined from this test alone.

Applying the Ratio Test to our given series \(1+\frac{2}{3}+\frac{3}{3^2}+\frac{4}{3^3}+\cdots\), the ratio of successive terms simplifies to \(\frac{n+1}{3n}\). As \(n\) approaches infinity, the limit of this ratio is \(\frac{1}{3}\), which is less than 1. Therefore, we can conclude that the series is convergent according to the Ratio Test.

Root Test

The Root Test is another tool to analyze the behavior of infinite series. Similar to the Ratio Test, the Root Test focuses on the \(n\)-th root of the absolute value of the \(n\)-th term. For a series \(\sum a_n\), the Root Test looks at the limit \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\).

The Root Test states:

• If \(L < 1\), the series converges.
• If \(L > 1\), the series diverges.
• If \(L = 1\), similar to the Ratio Test, the test does not provide a conclusive answer, and the series may converge or diverge.

While the Root Test was not used in the original exercise, understanding how it works and when it is best applied—typically in cases where terms incorporate \(n\)-th roots or powers—is important for a well-rounded comprehension of series convergence tests.

Sequences and Series

In mathematics, sequences and series are fundamental concepts. A sequence is an ordered list of numbers that often follow a specific rule, called the general term. An example of a sequence is the natural numbers \(1, 2, 3, \dots\). A series, on the other hand, is the sum of the terms of a sequence.

For example, the sum \(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots\) is a series derived from the sequence \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots\). Notably, for series to be meaningful, we often consider their behavior as the number of terms goes to infinity, which leads us to explore the concepts of convergence and divergence.

Convergent Series

A convergent series is one where the sum of its terms approaches a specific number, or limit, as more and more terms are added. This means that as you sum more terms of the series, the total sum gets closer and closer to a certain finite value, and it does not keep getting larger indefinitely.

Convergent series are significant because they allow us to operate with an infinite number of terms in a meaningful way. The series given in the exercise \(1+\frac{2}{3}+\frac{3}{3^2}+\frac{4}{3^3}+\cdots\) is an example of a convergent series. As we calculated earlier using the Ratio Test, it approaches a limit, which means that despite having infinitely many terms, their total sum doesn't increase beyond a certain point.