Problem 44 - A Graphical Approach to Precalculus with Limits

Question

CONCEPT CHECK The number 0.999 ... 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 infinite series is 1.

Step 1: Identify the Starting Term

The first term of the infinite geometric sequence is identified as \( a_1 = 0.9 \). This is given directly in the problem statement as the first term of the sequence.

Step 2: Determine the Common Ratio

The common ratio \( r \) is given as 0.1. This represents the factor by which we multiply each term to get the next term in the sequence, i.e., each term is 10% of the previous term.

Step 3: Use the Formula for Infinite Geometric Series Sum

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

Step 4: Substitute the Values into the Formula

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

Step 5: Calculate the Sum

Simplify the expression \( \frac{0.9}{1 - 0.1} \) to obtain \( \frac{0.9}{0.9} \). When calculated, this gives \( 1 \). Hence, the sum of the infinite series is 1.

Key Concepts

Common Ratio

In an infinite geometric series, the common ratio is a crucial element that determines the behavior of the sequence. It is the constant factor by which we multiply each term to obtain the next term. In our example with the sequence 0.9, 0.09, 0.009, ..., the common ratio \( r \) is 0.1. This means each subsequent term is 10% of the previous one.
Understanding the common ratio helps us know how rapidly the terms in the series decrease or increase. If \( |r| < 1 \), like our 0.1, the series terms get smaller, which is vital for convergence. Conversely, if \( |r| > 1 \) the terms would grow, potentially rendering the series divergent if it were infinite.
To find the common ratio in any geometric sequence, divide a term by its preceding term. For example, to confirm our common ratio, take the second term, 0.09, and divide it by the first term, 0.9. You will get \( r = 0.1 \). This constant growth factor is critical for the series' analysis and aids in determining the series' convergence.

Convergence of Series

Convergence is a vital concept when working with infinite series. An infinite geometric series converges if and only if the absolute value of the common ratio is less than 1, \( |r| < 1 \). This condition ensures that as you continue to add terms, the sum approaches a finite limit rather than growing infinitely.
In our example with the sequence 0.9, 0.09, 0.009, ..., the common ratio \( r = 0.1 \) is indeed less than 1. This tells us that the series will converge, meaning it will approach a specific value — in this case, 1. As more terms are added, each new term becomes less significant in altering the sum, eventually getting so small that their impact is negligible.
This property of convergence is what allows us to use the formula for the sum of an infinite geometric series. Without convergence, there would be no meaningful single value that the series approaches, making the sum undefined or infinite.

Sum of Series

The sum of an infinite geometric series can be determined by the formula \( S_\infty = \frac{a_1}{1 - r} \), where \( a_1 \) is the first term, and \( r \) is the common ratio. This formula is applicable only when the series converges, which means \( |r| < 1 \).
In our exercise, the starting term \( a_1 = 0.9 \) and the common ratio \( r = 0.1 \), both fit perfectly into this formula. Substituting these values gives us:
\[ S_\infty = \frac{0.9}{1 - 0.1} \]
This simplifies to \( \frac{0.9}{0.9} \), resulting in a sum of 1. This calculation shows that the infinite series 0.9, 0.09, 0.009, ... will precisely sum to 1 as the number of terms approaches infinity.

• The denominator \( 1 - r \) ensures the formula is valid only for converging series.
• The numerator \( a_1 \) ensures that the initial value contributes significantly to the sum.

Understanding how to calculate the sum is crucial in fields that use series for convergence, approximation, and various real-world applications.