Problem 23 - Thomas Calculus

Question

Which of the series in Exercises 13 46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{2^{n}}{3^{n}} $$

Step-by-Step Solution

Answer

The series converges because it is a geometric series with a ratio less than 1.

Step 1: Identify the Series Form

The given series is \(\sum_{n=1}^{\infty} \frac{2^n}{3^n}\). We can rewrite the general term as \( \left( \frac{2}{3} \right)^n \). Recognizing this form is important as it hints at specific convergence tests we may use.

Step 2: Recognize a Geometric Series

The series is a geometric series. In general, a geometric series is written as \(\sum_{n=1}^{\infty} ar^n \) (starting from \( n=1 \)) or \( ar^{n-1} \) (starting from \( n=0 \)). Here, the first term \( a \) is 1, and the common ratio \( r \) is \( \frac{2}{3} \).

Step 3: Apply Convergence Test for Geometric Series

A geometric series \( \sum_{n=0}^{\infty} ar^n \) converges if the absolute value of the common ratio \( |r| < 1 \). If \(|r| \geq 1\), it diverges. Here, \( |r| = \left| \frac{2}{3} \right| = \frac{2}{3} < 1 \). This condition is satisfied, meaning the series converges.

Step 4: Determine the Sum of the Series

For a converging geometric series \( \sum_{n=0}^{\infty} ar^n \), the sum is given by \( \frac{a}{1-r} \). Since our series starts from \( n=1 \), we adjust for the first few terms. The sum of the series is \( \frac{1}{1-\frac{2}{3}} = 3 \). Hence, \( \sum_{n=1}^{\infty} \left( \frac{2}{3} \right)^n = 3 - a = 3 - 1 = 2 \), which is the total sum starting from \( n=1 \).

Key Concepts

Geometric Series

A geometric series is a fascinating and straightforward concept in mathematics. It is a series where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in our original exercise, the series is \( \sum_{n=1}^{\infty} \frac{2^n}{3^n} \), which can be rewritten as \( \sum_{n=1}^{\infty} \left( \frac{2}{3} \right)^n \). Here, the common ratio \( r \) is \( \frac{2}{3} \).

When dealing with geometric series, there are some critical aspects to note:

• The first term \( a \) of the series in most cases influences the outcome of the sum. For the continuous sequence, it would be the value inside the sum formula.
• The common ratio \( r \) determines the behavior of the series; it dictates whether the series grows, shrinks, or remains steady.

Recognizing the structure of a geometric series helps in applying the correct convergence tests and finding the sum when the series converges.

Convergence Tests

Convergence tests are methods used to determine whether an infinite series converges or diverges. For geometric series, there's a handy test that involves the common ratio. The rule is simple: a geometric series \( \sum_{n=0}^{\infty} ar^n \) converges if the absolute value of the common ratio \( |r| < 1 \). If not, it diverges.

In the original exercise, the common ratio \( r = \frac{2}{3} \), and since \( |r| = \frac{2}{3} < 1 \), the series converges. This type of easy-to-use convergence test makes analyzing geometric series more approachable. Knowing when a series will sum to a finite value or continue growing indefinitely is crucial in both theoretical and practical applications.

While the geometric series has its convergence test, there are other types of tests for more complex series, such as the Ratio Test, Root Test, and Integral Test. Each of these has criteria to evaluate convergence and serves different types of series, making them versatile tools for analyzing infinite sums.

Infinite Series

Infinite series, as the name suggests, continue indefinitely, adding endless terms together. These series are of great interest in mathematics as they can represent complex functions and physical phenomena.

An infinite series can either converge or diverge:

• Convergent series: These are series where the sum of all terms settles to a finite number as more terms are added. For example, the geometric series in our exercise converges to a sum of 2.
• Divergent series: These series don’t settle to a finite sum but instead grow indefinitely.

Determining the behavior of an infinite series is essential in calculus and real-world applications. By using different tests, like the geometric series test, you can conclude about whether sums will result in finite or infinite values. Each type of infinite series and its convergence properties are a step into understanding complex mathematical and physical theories.