Problem 73 - 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. $$\sum_{n=1}^{\infty} 2\left(\frac{7}{3}\right)^{n-1}$$

Step-by-Step Solution

Answer

The sum of the series is not possible as it diverges to infinity.

Step 1: Identify the First Term and the Common Ratio

The first term, a, is 2 since \(2(7/3)^{1-1} = 2\). The common ratio, r, is 7/3 as it's the base of the power in the series.

Step 2: Check if the sum can be computed

The sum of an infinite geometric series can only be computed if the absolute value of the common ratio is less than 1 (i.e., \(|r| < 1\)). In this case, the common ratio is 7/3. Since \(|7/3| > 1\), this series does not meet the criteria for finding the sum.

Step 3: Arrive at Conclusion

Since the common ratio does not meet the criteria, we conclude the sum of the series is not possible. Instead, it diverges to infinity.

Key Concepts

Sum of an Infinite Geometric Series

Understanding the sum of an infinite geometric series is crucial for working with series in mathematics. An infinite geometric series is a sequence 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. The sum of such a series can be found using the formula:
\[\begin{equation}S = \frac{a}{1 - r}, \ \end{equation}\] where 'S' represents the sum, 'a' is the first term, and 'r' is the common ratio. This formula is applicable only when the absolute value of the common ratio is less than 1 (\[\begin{equation}|r| < 1\end{equation}\]). If the common ratio is greater or equal to 1, the terms of the series would grow larger indefinitely, and the series would not converge to a finite sum. In simple terms, the series would just keep on adding to infinity without ever reaching a specific value.

Common Ratio

The common ratio in a geometric series is the constant factor between consecutive terms. For the series to converge, that is, to add up to a finite sum, this common ratio must be between -1 and 1, not inclusive. The smaller the absolute value of the common ratio, the faster the series terms decrease in size, leading to quicker convergence. It's a measure of how quickly the 'jump' from one term to the next occurs, being a shrinkage if the value is between -1 and 1, or a growth if otherwise.

Geometric Series Convergence

A geometric series converges when the sum of its terms approaches a finite limit. For this to occur, the series must have a common ratio with an absolute value less than 1. When a geometric series converges, it's possible to calculate the sum of all its terms, even though they are infinitely many, because the terms get smaller and add up to a certain value. In practical terms, if you were to keep adding the terms of a convergent geometric series forever, you'd end up approaching closer and closer to a specific number, which constitutes the sum of the series.

Series Divergence

On the other hand, series divergence is when the terms of a series do not approach a finite limit as they are added. This happens when the common ratio's absolute value is equal to or greater than 1. In such cases, the series terms either grow without bound or oscillate without settling down. A divergent series means that you cannot sum it up in traditional sense. Instead of approaching a specific value, the sum would either go to infinity or just keep bouncing around endlessly, much like an endless game of ping-pong where the ball never stops.