Problem 42 - Algebra 2

Question

a. A classmate uses the formula for the sum of an infinite geometric series to evaluate \(1+1.1+1.21+1.331+\ldots\) and gets \(-10 .\) Is your classmate's answer reasonable? Explain. b. Error Analysis What did your classmate fail to check before using the formula?

Step-by-Step Solution

Answer

No, the classmate's answer is not reasonable as the sum of the series is not negative. The classmate failed to confirm if the common ratio is less than 1, which is a prerequisite for the formula used.

Step 1: Identify the geometric series parameters

First, identify the parameters of the geometric series. The first term (a) is 1, and the common ratio (r) between subsequent terms is 1.1.

Step 2: Check the validity of the geometric series sum formula

The sum of an infinite geometric series is given by the formula S = a / (1 - r), valid when the absolute value of r is less than 1. However, in this series, r (1.1) is greater than 1, meaning the sum cannot be computed following this formula as the series is diverging.

Step 3: Verify the sum

Given that r > 1 and the first term a > 0, the summation should result in a positive and growing sum, not in a negative value. Therefore, the classmate's answer (-10) is not reasonable.

Step 4: Error analysis

The classmate failed to ensure that the geometric series was converging (i.e., |r| < 1) before applying the formula for the sum of an infinite geometric series. This led to an incorrect and unreasonable result.

Key Concepts

Convergent Series

In mathematics, a convergent series is a series where the sum of its infinite sequence approaches a specific finite number. Convergence is a crucial aspect in determining whether a series has a valid sum. When we talk about convergence for geometric series, we particularly look at the common ratio. A series converges if the absolute value of the common ratio, denoted as \(|r|\), is less than 1. This is because each subsequent term becomes smaller and shrinks towards zero.

In the context of the infinite geometric series, if the series converges, we can use the formula \(S = \frac{a}{1 - r}\) to calculate its sum, where \(a\) is the first term, and \(r\) is the common ratio.

However, if the series does not converge—meaning \(|r| \geq 1\)—then the terms do not approach zero, and the sum cannot be determined using this formula. In the original exercise, the series does not meet the convergence criteria, as the common ratio is 1.1, which is greater than 1.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio, \(r\). For example, in the sequence \(1, 1.1, 1.21, 1.331, \ldots\), each term is derived by multiplying the preceding term by \(1.1\).

A geometric sequence can be finite or infinite. An infinite geometric sequence is simply one that continues indefinitely without reaching a stopping point.

Understanding geometric sequences is fundamental for solving problems involving the sum of an infinite geometric series, as identifying the first term \(a\) and the common ratio \(r\) is essential.

• The formula for a geometric sequence is given by \( a_n = a \cdot r^{n-1} \), where \(a_n\) is the \(n\)-th term.
• This recursive relationship confirms how each term is formed based on the consistent multiplication by the common ratio.

In relation to the exercise, identifying the correct first term and common ratio is key to evaluating the series.

Common Ratio

The common ratio in a geometric sequence is the factor between any two consecutive terms. It denotes how much each term is multiplied to arrive at the next term in the sequence. This can be calculated by dividing any term by its previous term. For instance, in the sequence from the exercise \(1, 1.1, 1.21, 1.331, \ldots\), the common ratio \(r\) is 1.1.

The common ratio greatly impacts whether a geometric series will converge or diverge. Specifically:

• If \(|r| < 1\), the series converges, and we can calculate its sum using specific formulas.
• If \(|r| \geq 1\), the series diverges, making the sum infinite or undefined.

In this exercise, the misstep occurred because the assumed convergence did not apply, as the common ratio was 1.1, leading to an infinite and diverging outcome.

Error Analysis

Error analysis is a critical process in problem-solving and mathematics, where we identify mistakes in calculations or logical reasoning. In this exercise, the error analysis reveals where the classmate went wrong.

First, it's essential to understand the conditions under which the formula for the sum of an infinite geometric series can be applied. The formula \(S = \frac{a}{1 - r}\) is only valid if the absolute value of the common ratio \(|r|\) is less than 1.

• Step 1 should always entail verifying the common ratio since values equal to or exceeding 1 result in a lack of convergence.
• Applying a formula without checking assumptions is a common mistake, which can lead to incorrect conclusions, such as an unfeasible negative sum like -10 in a diverging series.

This mistake emphasizes the importance of thoroughly checking all preconditions before proceeding with any calculations or applying formulas.