Problem 40 - A Graphical Approach to College Algebra

Question

The number \(0.999 \ldots\) can be written as the sum of the terms of an infinite geometric sequence: \(0.9+0.09+0.009+\cdots\) Here we have \(a_{1}=0.9\) and \(r=0.1 .\) Use the formula for \(S_{\infty}\) to find this sum.

Step-by-Step Solution

Answer

The sum of the series is 1.

Step 1: Identify the first term and common ratio

In the given geometric sequence, the first term \(a_1\) is 0.9 and the common ratio \(r\) is 0.1. These values are essential for applying the formula for the sum of an infinite geometric series.

Step 2: Recall the formula for the sum of an infinite geometric series

The formula for the sum \(S_{\infty}\) of an infinite geometric series is given by \(S_{\infty} = \frac{a_1}{1 - r}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Step 3: Substitute the values into the formula

Substitute \(a_1 = 0.9\) and \(r = 0.1\) into the sum formula: \[ S_{\infty} = \frac{0.9}{1 - 0.1} \]

Step 4: Calculate the denominator

Calculate the denominator \(1 - 0.1\) to get 0.9. The expression now becomes: \[ S_{\infty} = \frac{0.9}{0.9} \]

Step 5: Perform the division

Divide 0.9 by 0.9 to find the sum of the series: \[ S_{\infty} = 1 \]

Key Concepts

Geometric Sequence

A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In our example, we are dealing with the sequence:

• 0.9
• 0.09
• 0.009
• ...

Each of these terms is derived by multiplying the previous term by 0.1, the common ratio. This sequence's progressive diminishing is a hallmark of an infinite geometric sequence when the common ratio is less than one. You can easily spot a geometric sequence by checking the division of any term by its predecessor. If this ratio remains constant, you have a geometric sequence at hand. In this infinite sequence, although the terms keep getting smaller, they still add up to a particular number. This feature is unique and will be further explored in the concept of the sum of the series.

Sum of Series

The sum of an infinite geometric series is the value that the series converges to as more and more terms are added. This is different from finite series where the sum is simply the addition of a certain number of terms. In an infinite geometric series, if the common ratio's absolute value is less than one, the series will converge, and you can find the sum using a specific formula. For our sequence, the sum is computed using the formula: \[ S_{\infty} = \frac{a_1}{1 - r} \] where \(a_1\) is the first term and \(r\) is the common ratio. Our exercise shows that for \(a_1 = 0.9\) and \(r = 0.1\), substituting into the formula gives: \[ S_{\infty} = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1 \] Hence, the sum of this series is 1, meaning despite the infinite number of terms, they practically sum to 1. This reveals the fascinating property of infinite series where the total can be a finite value.

Common Ratio

The common ratio is the constant factor between consecutive terms of a geometric sequence. It plays a critical role in defining the behavior of a series, especially in determining whether and how the series converges. In our case, the common ratio is 0.1. Properties of the Common Ratio:

• If the common ratio's absolute value is less than one, the series converges, meaning its sum approaches a finite limit.
• If the ratio is greater than one, the series diverges, implying the sum increases indefinitely.
• In our exercise, the common ratio of 0.1, being less than one, ensures that the infinite series converges to a finite sum of 1.

The common ratio is not just a number; it carries significant weight in determining the destiny of a geometric series. In calculations or real-world scenarios, checking the common ratio first can hint at whether your series is heading towards a sensible, finite outcome or an infinite progression.