Problem 13 - 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.) $$ \sum_{n=1}^{\infty} \frac{1}{10^{n}} $$

Step-by-Step Solution

Answer

The series converges, with a sum of \(\frac{1}{9}\).

Step 1: Identify the Series Type

The given series is \( \sum_{n=1}^{\infty} \frac{1}{10^{n}} \). Recognize this as a geometric series, where each term is of the form \( ar^n \) with \( a = \frac{1}{10} \) and \( r = \frac{1}{10} \).

Step 2: Check the Common Ratio

In a geometric series \( \sum a r^n \), convergence is determined by the absolute value of the common ratio \( r \). If \(|r| < 1\), the series converges. For this series, \(|\frac{1}{10}| = 0.1 < 1\).

Step 3: Apply the Geometric Series Test

Since \(|r| = 0.1 < 1\), the series converges. According to the geometric series formula, the sum of an infinite geometric series is \( \frac{a}{1-r} \).

Step 4: Calculate the Sum

Use the formula for the sum of an infinite geometric series \( \frac{a}{1-r} \), where \( a = \frac{1}{10} \) and \( r = \frac{1}{10} \). Compute the sum as \( \frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{\frac{1}{10}}{\frac{9}{10}} = \frac{1}{9} \).

Key Concepts

Series Convergence

Understanding whether a series converges or diverges is crucial in calculus. A series is a sum of terms of a sequence. In the context of the geometric series, convergence is determined by the common ratio. For a series to converge, the sequence's terms should approach zero as they progress towards infinity. This means that the series will have a finite sum. If a series converges, you can calculate its sum. However, if it diverges, the sum will go off to infinity. The rule of convergence for a geometric series is specifically tied to the value of its common ratio. If we know our series is geometric, checking convergence often simply involves calculating the common ratio.

Common Ratio

The common ratio is a vital element in a geometric series. It is the factor that each term is multiplied by to get the next term in the series. To find the common ratio, you divide any term in the sequence by its preceding term. In the series \( \sum_{n=1}^{\infty} \frac{1}{10^{n}} \) given in the exercise, the common ratio \( r \) is \( \frac{1}{10} \). Determining whether the series converges depends significantly on this value.

• If \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series diverges.

In our series, since \( |\frac{1}{10}| = 0.1 \) which is less than 1, the series converges.

Infinite Series

An infinite series is a series that continues indefinitely, meaning it has an infinite number of terms. These types of series can sometimes have a finite sum, depending on their convergence. Infinite series are essential in mathematical calculations, as they help model various phenomena in the real world. The series \( \sum_{n=1}^{\infty} \frac{1}{10^{n}} \) is an example of an infinite series. Despite having an endless number of terms, this particular series converges, meaning it sums up to a finite number. The concept might seem counterintuitive, but with specific conditions on the common ratio, it is achievable.

Geometric Series Test

The geometric series test is a handy tool for quickly determining the convergence of a geometric series. It essentially states that a geometric series \( \sum ar^n \) converges if the absolute value of the common ratio is less than 1. Otherwise, it diverges. This test cuts down the complexity of checking each series term because once you find the common ratio, the decision is straightforward. For our series \( \sum_{n=1}^{\infty} \frac{1}{10^{n}} \), performing the geometric series test reveals the series' convergence since \( |r| = 0.1 < 1 \). Hence, using the formula \( \frac{a}{1-r} \), where \( a \) is \( \frac{1}{10} \) and \( r \) is also \( \frac{1}{10} \), the sum of this infinite series is \( \frac{1}{9} \). This confirms that our understanding and application of the geometric series test is indeed correct.