Problem 57 - Thomas Calculus

Question

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=1}^{\infty} \frac{2}{10^{n}} $$

Step-by-Step Solution

Answer

The series converges with a sum of \(\frac{2}{9}\).

Step 1: Identify the Series Type

Observe that the series is given by \( \sum_{n=1}^{\infty} \frac{2}{10^{n}} \). This is a geometric series with the first term \( a = \frac{2}{10^1} = 0.2 \) and a common ratio \( r = \frac{1}{10} \).

Step 2: Analyze the Convergence Criterion for Geometric Series

Recall that a geometric series \( \sum_{n=0}^{\infty} ar^n \) converges if the absolute value of the common ratio \( |r| < 1 \). Here, \( |r| = \frac{1}{10} = 0.1 \), which is less than 1.

Step 3: Confirm and Conclude Convergence

Since \( |r| = 0.1 < 1 \), the series \( \sum_{n=1}^{\infty} \frac{2}{10^{n}} \) converges.

Step 4: Calculate the Sum of the Convergent Series

For a convergent geometric series where the first term \( a = 0.2 \) and the common ratio \( r = 0.1 \), the sum \( S \) can be found using the formula: \[ S = \frac{a}{1 - r} \]. Substitute the values to get: \[ S = \frac{0.2}{1 - 0.1} = \frac{0.2}{0.9} = \frac{2}{9} \].

Key Concepts

Convergence in Series

Convergence is a key concept when analyzing series, particularly when considering whether a series converges or diverges. When we say a series converges, it means the sum of its infinite terms approaches a particular finite value. This is important in mathematics as not all series will add up to a finite number.

For a geometric series, such as the one in our example, convergence is determined by its common ratio ("r"). Specifically, the series converges if the absolute value of this ratio (|r|) is less than 1. Let's consider the series \( \sum_{n=1}^{\infty} \frac{2}{10^{n}} \). Here, the common ratio is 0.1, which is indeed less than 1, thus confirming the series' convergence. This means that as we sum the terms from 1 to infinity, they approach a specific finite value. Understanding why a series converges is crucial for solving countless mathematical problems and applications.

Understanding the Common Ratio

The common ratio in a geometric series is the constant factor that you multiply by each term to get to the next term. This ratio plays a crucial role in determining whether the series converges or diverges.

Consider a geometric series given by the terms \(a, ar, ar^2, ar^3, \ldots\). Here, "a" is the first term, and "r" is the common ratio. For the example series \( \sum_{n=1}^{\infty} \frac{2}{10^{n}}\), the common ratio ("r") is 0.1, calculated by dividing any term by its preceding term. It is also important to note that for convergence, the absolute value \(|r|\) must be less than 1. This ensures that as you progress through the terms of the series, each gets progressively smaller, leading to a finite sum.

Knowing the common ratio helps not only clarify how the series behaves in terms of convergence but also in constructing the formula to find the sum of the series.

Calculating the Sum of a Series

Once we've determined that a series converges, like our geometric series, we can find its sum. The formula for the sum "S" of an infinite geometric series with first term "a" and common ratio "r" is:

• \[ S = \frac{a}{1 - r} \]

For the example series\( \sum_{n=1}^{\infty} \frac{2}{10^{n}} \), we've already identified that "a" is 0.2 (the first term), and "r" is 0.1. By plugging these values into the sum formula, we calculate:

• \[ S = \frac{0.2}{1 - 0.1} = \frac{0.2}{0.9} \]
• \( S = \frac{2}{9} \)

This sum represents the value that the series approaches as we add an infinite number of terms. Calculating this sum not only confirms convergence but also provides a useful numerical answer to the problem posed by the original geometric series.