Problem 104 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a formula to find the sum of the infinite geometric series \(3+1+\frac{1}{3}+\frac{1}{9}+\cdots\) and then checked my answer by actually adding all the terms.

Step-by-Step Solution

Answer

The usage of the formula to find the sum of the infinite geometric series makes sense and it gives 4.5. However, the attempt to check the answer by direct addition doesn't make sense because it involves physically summing an infinite number of terms, which is impossible.

Step 1: Identify The Geometric Series

The given series \(3+1+\frac{1}{3}+\frac{1}{9}+\cdots\) is a geometric series, because each term is a fixed multiple of the term before it. Here, the ratio, denoted by 'r', between successive terms is \(1/3\). The first term, denoted by 'a', is 3.

Step 2: Establish the Formula for the Sum of an Infinite Geometric Series

The sum, 'S', of an infinite geometric series is given by the formula \(S = \frac{a}{1 - r}\) if the absolute value of r is less than 1. If |r| is greater than or equal to 1, the series diverges and does not have a sum.

Step 3: Apply the Formula to the Given Series

Given that a = 3, and r = 1/3. By substituting these values into the formula we get: \(S = \frac{3}{1 - 1/3} = \frac{3}{2/3} = 4.5\). Hence, the sum of the infinite geometric series is 4.5.

Step 4: Analyze the Practicality of Checking by Addition

The second part of the statement suggests summing the infinite series by direct addition of every term which is practically impossible. One cannot physically add infinite terms. That’s why the statement does not make sense.

Key Concepts

Sum of Geometric Series

Understanding the sum of a geometric series can often be counterintuitive, as it involves summing up an infinite number of terms. However, through a concept known as convergence, mathematicians have developed a way to find this sum under certain conditions.

For an infinite geometric series, if the common ratio between consecutive terms, usually denoted as 'r', is between -1 and 1 (excluding these), then the series converges to a finite value. The sum 'S' can be found using the formula \(S = \frac{a}{1 - r}\), where 'a' is the first term of the series. When |r| is not less than 1, the series does not converge, hence a sum cannot be calculated. This is crucial for our sense-making in mathematics, as it tells us that an infinite sum can indeed have a finite result when handled with proper mathematical tools.

Convergence of Series

The concept of convergence is vital in understanding infinite series. Simply put, a series converges if its terms approach a specific value as you add more and more terms. In the context of a geometric series, if the absolute value of the common ratio 'r' is less than 1, it means each subsequent term is getting proportionally smaller, and the series adds up to a finite number.

To visualize convergence, consider a series as a process of adding slices of a pie, where each slice is smaller than the previous one. If you start with half a pie and each slice is half the size of the one before, you will eventually come close to having a whole pie without ever exceeding it. This is similar to how an infinite geometric series with |r|<1 behaves.

Geometric Series Formula

The geometric series formula is a powerhouse tool that can, under the right conditions, summarize an otherwise incomprehensible infinity of terms into a single, comprehensible sum. For the infinite series, this formula is given by \(S = \frac{a}{1 - r}\), where 'S' is the sum of the series, 'a' is the first term, and 'r' is the common ratio.

Application:

To apply this formula, we identify 'a' and 'r' from the series. For the series in our exercise, 'a' is 3 and 'r' is 1/3. Plugging these into the formula, we get our sum, \(S = \frac{3}{1 - 1/3} = \frac{3}{2/3} = 4.5\). The beauty of this formula lies in its simplicity and power to unleash the finite from the clutches of the infinite.

Mathematical Reasoning

Mathematical reasoning is what enables us to validate whether certain statements about mathematical concepts hold true. It involves logical thinking, pattern recognition, and an understanding of mathematical laws and formulas. When one claims to have used a formula to find the sum of an infinite series and then checked the result by adding all terms, we rely on mathematical reasoning to identify the flaw.

In our scenario, it is logically implausible to 'add all terms' of an infinite series as infinity does not constitute a quantity that can be physically manipulated or reached. Hence, mathematical reasoning tells us that although the first part of the statement about using a formula makes sense, the latter part does not, as it defies the very nature of infinity.