Problem 98 - Precalculus with Limits

Question

In Exercises 93 - 106, find the sum of the infinite geometric series. \( \sum_{n=0}^{\infty}\left(\dfrac{1}{10}\right)^n \)

Step-by-Step Solution

Answer

The sum of the infinite geometric series is \( \frac{10}{9} \)

Step 1: Identification of 'a' and 'r'

In this geometric series, 'a', the first term, is equal to 1, due to the n=0 for the first term (since \( \left(\frac{1}{10}\right)^0 = 1)\). The common ratio 'r' is equal to \( \frac{1}{10} \), as each term is multiplied by this factor.

Step 2: Application of the Formula

Now, we can apply the formula \( S = \frac{a}{1 - r} \) to find the sum of the series. Substituting the given values, we get \( S = \frac{1}{1 - \frac{1}{10}} \).

Step 3: Solve for S

By simplifying the equation on the right-hand side, the sum S will be calculated: \(S = \frac{1}{\frac{9}{10}} = \frac{10}{9}\).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For instance, in the series 2, 6, 18, 54, ..., the common ratio is 3 because each term is three times the term before it. Understanding the role of the first term, denoted as 'a', and the common ratio 'r', is crucial when working with geometric series.

In the given exercise, the first term 'a' is 1, and the ratio 'r' is \( \frac{1}{10} \), resulting in a series that gets progressively smaller. This decreasing nature is typical of a geometric series with a ratio between -1 and 1. It's this attribute that allows us to calculate the sum of an infinite number of terms, assuming the series is convergent.

Convergence of Series

The convergence of a series is a fundamental concept that determines whether the sum of its terms approaches a finite number as the number of terms increases indefinitely. For a geometric series, convergence occurs if and only if the absolute value of the common ratio 'r' is less than 1. In mathematical terms, a geometric series is convergent when \( |r| < 1 \).

With a common ratio of \( \frac{1}{10} \), the series in the exercise definitely converges because the ratio is well within the bounds of convergence. This is because each subsequent term is smaller than the previous term by a factor of ten, leading to a diminishing addition to the sum. It's important to note that this condition is what makes it possible to use the formula \( S = \frac{a}{1 - r} \) to find the sum of an infinite geometric series.

Infinite Series

An infinite series is the sum of all terms in a sequence that goes on indefinitely. The idea might sound perplexing at first—how can we talk about the sum of an infinite number of things? The beauty of mathematics lies in its ability to make sense of such concepts through convergence. If a series converges, then it has a specific finite sum, even though it has an infinite number of terms.

For the geometric series in our exercise, the sum of an infinite number of terms can be found using the formula given since the series is convergent. We calculated the sum to be \( \frac{10}{9} \), demonstrating that even though the series continues indefinitely, the total sum remains within the realm of finite numbers, a remarkable and counterintuitive result that highlights the elegance of mathematical principles in describing infinite processes.