Series Convergence: Does ∑ (3/4)(2)^n Converge Or Diverge?

Understanding whether a series converges or diverges is a fundamental concept in calculus and mathematical analysis. In this comprehensive exploration, we'll dive deep into the series ∑(from n=1 to ∞) (3/4)(2)^n, dissecting its components and applying the necessary tests to determine its behavior. Our primary goal is to not only identify whether this series converges or diverges but also to provide a clear and intuitive explanation behind the result. Whether you're a student grappling with series convergence, a seasoned mathematician looking for a refresher, or simply curious about the intricacies of infinite sums, this article will equip you with the knowledge and understanding you need.

Understanding the Series ∑(from n=1 to ∞) (3/4)(2)^n

To begin our analysis, let's first break down the series in question: ∑(from n=1 to ∞) (3/4)(2)^n. This notation represents an infinite sum, where each term is generated by the expression (3/4)(2)^n, and 'n' takes on integer values starting from 1 and extending to infinity. In simpler terms, we are adding up an infinite number of terms, each of which is obtained by multiplying (3/4) with 2 raised to the power of 'n'.

The first few terms of this series can be calculated as follows:

• When n = 1: (3/4)(2)^1 = (3/4) * 2 = 3/2
• When n = 2: (3/4)(2)^2 = (3/4) * 4 = 3
• When n = 3: (3/4)(2)^3 = (3/4) * 8 = 6
• When n = 4: (3/4)(2)^4 = (3/4) * 16 = 12

and so on. As you can see, the terms of the series are 3/2, 3, 6, 12, and so on. The key observation here is that these terms are growing larger and larger as 'n' increases. This is a crucial clue in determining whether the series converges or diverges. A series converges if the sum of its terms approaches a finite value as the number of terms increases infinitely. Conversely, a series diverges if the sum of its terms does not approach a finite value; instead, it either grows without bound or oscillates indefinitely.

To further classify this series, we can recognize it as a geometric series. A geometric series is a series where each term is multiplied by a constant ratio to obtain the next term. In our case, the constant ratio, often denoted as 'r', is 2. We can see this by dividing any term by its preceding term: for instance, 3 / (3/2) = 2, 6 / 3 = 2, and so on. The general form of a geometric series is ∑(from n=1 to ∞) ar^(n-1), where 'a' is the first term and 'r' is the common ratio. Understanding this structure is vital because the convergence or divergence of a geometric series depends entirely on the value of its common ratio 'r'.

Convergence and Divergence Tests: The Geometric Series Test

Now that we've identified our series as a geometric series with a common ratio of 2, we can leverage a powerful tool known as the Geometric Series Test to determine its convergence behavior. The Geometric Series Test provides a straightforward criterion for determining whether a geometric series converges or diverges based solely on the value of its common ratio 'r'.

The test states the following:

• If |r| < 1 (the absolute value of 'r' is less than 1), then the geometric series converges, and its sum can be calculated using the formula S = a / (1 - r), where 'S' is the sum, 'a' is the first term, and 'r' is the common ratio.
• If |r| ≥ 1 (the absolute value of 'r' is greater than or equal to 1), then the geometric series diverges.

This test is incredibly useful because it provides a quick and decisive way to analyze the behavior of geometric series without having to resort to more complex convergence tests. The intuition behind this test lies in the behavior of the terms as 'n' approaches infinity. When |r| < 1, the terms of the series become progressively smaller, approaching zero as 'n' increases. This allows the sum to converge to a finite value. However, when |r| ≥ 1, the terms either remain constant in magnitude or grow larger, preventing the sum from settling on a finite value, hence the divergence.

In our specific case, the series ∑(from n=1 to ∞) (3/4)(2)^n has a common ratio r = 2. The absolute value of 2 is |2| = 2, which is clearly greater than 1. Therefore, according to the Geometric Series Test, this series diverges. This means that as we add more and more terms of this series, the sum does not approach a finite value; instead, it grows without bound. This conclusion aligns with our earlier observation that the terms of the series are increasing in magnitude, indicating a divergent behavior.

It's important to emphasize that the Geometric Series Test is a specific test designed for geometric series. While there are other convergence tests available, such as the Ratio Test, Root Test, and Comparison Tests, the Geometric Series Test offers the most direct and efficient method for analyzing geometric series. Understanding and applying this test correctly is crucial for anyone studying series and sequences in calculus and mathematical analysis.

Applying the Geometric Series Test to ∑(from n=1 to ∞) (3/4)(2)^n

Let's now formally apply the Geometric Series Test to our series, ∑(from n=1 to ∞) (3/4)(2)^n, to solidify our understanding and arrive at a definitive conclusion. As we've already established, this series is a geometric series with the first term a = 3/2 (when n = 1) and a common ratio r = 2. The Geometric Series Test tells us that if |r| < 1, the series converges, and if |r| ≥ 1, the series diverges.

In our case, the common ratio r is 2, and its absolute value is |2| = 2. Since 2 is greater than 1, we have |r| ≥ 1. Therefore, according to the Geometric Series Test, the series ∑(from n=1 to ∞) (3/4)(2)^n diverges. This divergence means that the sum of the terms in the series does not approach a finite value as we add more and more terms. Instead, the sum grows without bound, tending towards infinity.

To further illustrate this divergence, let's consider the partial sums of the series. The partial sums are the sums of the first 'k' terms of the series, where 'k' is a positive integer. Let's calculate the first few partial sums:

• S1 (sum of the first term) = 3/2 = 1.5
• S2 (sum of the first two terms) = 3/2 + 3 = 9/2 = 4.5
• S3 (sum of the first three terms) = 3/2 + 3 + 6 = 21/2 = 10.5
• S4 (sum of the first four terms) = 3/2 + 3 + 6 + 12 = 45/2 = 22.5

As you can observe, the partial sums are increasing rapidly. This trend continues as we add more terms, with the partial sums growing larger and larger without approaching any specific finite value. This behavior is a clear indication of divergence.

The Geometric Series Test provides a concise and powerful method for determining the convergence or divergence of geometric series. By simply examining the common ratio 'r', we can quickly ascertain the series's behavior. In the case of ∑(from n=1 to ∞) (3/4)(2)^n, the common ratio of 2 unequivocally leads to the conclusion that the series diverges.

Conclusion: The Series Diverges

In conclusion, through our detailed analysis and application of the Geometric Series Test, we have conclusively determined that the series ∑(from n=1 to ∞) (3/4)(2)^n diverges. This series, characterized by its exponentially increasing terms, fails to converge to a finite sum. The common ratio of 2, being greater than 1, serves as the key indicator of this divergent behavior.

Understanding the concept of series convergence and divergence is paramount in various branches of mathematics, including calculus, real analysis, and complex analysis. The Geometric Series Test, in particular, stands as a fundamental tool for analyzing geometric series, offering a straightforward criterion based on the common ratio. By mastering this test and its underlying principles, students and practitioners alike can effectively assess the behavior of geometric series and apply this knowledge to solve a wide range of problems.

This exploration of ∑(from n=1 to ∞) (3/4)(2)^n serves as a valuable illustration of the divergence phenomenon in series. It underscores the importance of careful analysis and the application of appropriate convergence tests to accurately determine the behavior of infinite sums. As you continue your mathematical journey, remember that a solid grasp of series convergence and divergence will serve as a cornerstone for tackling more advanced concepts and applications.

For further exploration of series and sequences, consider visiting Khan Academy's Calculus section on Series.