Problem 21 - Algebra 2

Question

Evaluate each infinite geometric series. $$ 1.1-0.11+0.011-\ldots $$

Step-by-Step Solution

Answer

The sum of the given infinite series is 1.0.

Step 1: Identify the first term and the common ratio

In the given series 1.1 - 0.11 + 0.011 - ..., the first term \(a\) is 1.1 and the common ratio \(r\) is -0.11/1.1 = -0.1. Since the absolute value of the ratio (-0.1) is less than 1, the series will converge, and we can proceed to evaluate it.

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

The sum \(S\) of an infinite geometric series with first term \(a\) and ratio \(r\) is given by the formula \(S = a/(1-r)\). Substituting \(a = 1.1\) and \(r = -0.1\) into this formula, we get \(S = 1.1/(1 - (-0.1)) = 1.1/1.1 = 1.0\).

Step 3: Confirm the answer

Thus, the sum of the given infinite geometric series is 1.0. This is a finite number, which confirms our determination that the series converges.

Key Concepts

Convergent Series

An infinite geometric series is a sum of terms that continue indefinitely. A series is called a *convergent series* if the sum of its infinite terms approaches a finite limit. This means the series gets closer and closer to a certain value, without exceeding it as more terms are added.

In the case of an infinite geometric series, convergence depends on the *common ratio* \(r\). Specifically, if the absolute value of \(r\) is less than 1 (\(|r| < 1\)), the series will converge. This is because each subsequent term becomes smaller, causing the entire sum to stabilize at a finite number.

For example, in the series

• 1.1 - 0.11 + 0.011 - 0.0011 + ...

the terms decrease in magnitude because the common ratio (-0.1) is less than 1 in absolute value. Thus, this series converges.

Common Ratio

The *common ratio* in a geometric sequence or series is the factor by which each term is multiplied to get the next term. It is a critical factor in determining the behavior of the series.

To calculate the common ratio \(r\), divide any term by the previous term. For the series given as

• 1.1 - 0.11 + 0.011 - ...

the common ratio is \(-0.1\). This is found by computing \(-0.11/1.1 = -0.1\).

The common ratio tells us:

• If \(|r| < 1\), the series will converge.
• If \(|r| = 1\), the series neither converges nor diverges in a typical manner.
• If \(|r| > 1\), the series will diverge.

This means for the series to stabilize at a certain value, each term should reduce in size, which only happens when the common ratio's absolute value is less than 1.

Sum of Series

The *sum of an infinite geometric series* can be calculated using a specific formula when the series is convergent. If the initial term is \(a\) and the common ratio is \(r\) (where \(|r| < 1\)), the sum \(S\) of the series is given by:\[S = \frac{a}{1-r}\]

In our example, the first term \(a = 1.1\) and the common ratio \(r = -0.1\). By substituting into the formula, we get:\[S = \frac{1.1}{1 - (-0.1)} = \frac{1.1}{1.1} = 1.0\]

This formula shows that even an infinite number of terms can sum to a finite value owing to convergence. It simplifies the process of evaluating seemingly complex sequences by providing a simple expression to determine the sum.

Geometric Sequence

A *geometric sequence* is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the *common ratio*.

Geometric sequences have unique properties:

• Each term can be expressed as \(a \times r^n\), where \(n\) is the term index starting from 0.
• They form the basis of geometric series, where the sequence terms are summed.

For the sequence starting with 1.1, it progresses as:

• 1.1, -0.11, 0.011, -0.0011, ...

Here, the *common ratio* of -0.1 applies to each successive term, dictating how they are calculated and how the series ultimately behaves in terms of convergence.