Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty} 6\left(\frac{2}{3}\right)^{n}$$

Step-by-Step Solution

Answer

The sum of the given infinite geometric series is 18.

Step 1: Identify the first term and the common ratio

First identify the first term \( a \) and the common ratio \( r \) in the series. In the given series \( \sum_{n=0}^{\infty} 6\left(\frac{2}{3}\right)^{n} \), we see that \( a = 6 \) and \( r = \frac{2}{3} \).

Step 2: Check the convergence of the series

To check whether the geometric series converges, we must ensure that the value of \( |r| < 1 \). In our case, \( |r| = |\frac{2}{3}| = \frac{2}{3} \), which is indeed less than 1. Therefore, the series converges.

Step 3: Calculate the sum of the series

Now that we know that the series converges, we can calculate its sum using the formula \( S = \frac{a}{1 - r} \). Substituting our values we get \( S = \frac{6}{1 - \frac{2}{3}} \)

Step 4: Simplify the result

Finally, simplify the expression to find the sum. This provides the final solution: \( S = \frac{6}{1 - \frac{2}{3}} = \frac{6}{\frac{1}{3}} = 6 \times 3 = 18 \).

Key Concepts

Convergence of Series

When discussing the concept of an infinite series, a pivotal question is whether the series converges or diverges. Convergence refers to the series' tendency to approach a specific value as the number of terms grows indefinitely. For a geometric series, this hinges on the value of the common ratio, denoted as 'r'. If the absolute value of 'r' is less than 1, that is, if \( |r| < 1 \), the series will converge. Conversely, if \( |r| \geq 1 \), the series will not have a finite sum and is said to diverge.

Considering the given problem, we identify \( r = \frac{2}{3} \), and since this value is less than 1, we conclude that the series converges. It's essential to recognize this property because only convergent series have a finite sum that we can calculate.

Geometric Series Sum

For a convergent infinite geometric series, we can find the sum using a specific formula: \( S = \frac{a}{1 - r} \), where 'S' is the sum, 'a' is the first term, and 'r' is the common ratio. After establishing that the series converges, the next step is to plug in the identified values into the formula. It's crucial to remember that this formula only applies if the series is convergent; that is, the condition \( |r| < 1 \) is satisfied.

In our exercise, with \( a = 6 \) and \( r = \frac{2}{3} \), the formula gives us \( S = \frac{6}{1 - \frac{2}{3}} \). Simplifying this, we find the sum to be 18. This sum is the limit the series approaches as the number of terms increases indefinitely. To put it simply, if you keep adding terms of the series forever, the total would approach 18.

Common Ratio

The common ratio 'r' in a geometric series is a fixed constant that each term is multiplied by to obtain the next term. For the series sum to be finite (i.e., for the series to converge), the common ratio must be between -1 and 1, excluding those endpoints. The magnitude of 'r' influences how quickly the terms of the series grow smaller; the closer the value of 'r' is to 0, the faster the decrease in the term sizes.

In the series given in the exercise, the common ratio is \( r = \frac{2}{3} \), which is between -1 and 1. This is not only crucial for convergence but also affects how we calculate the sum. A positive common ratio like ours indicates that the terms of the series decrease steadily towards zero, which aligns with the convergence we previously established.