Calculating The Sum Of The Infinite Geometric Series 0.2 - 0.026 + 0.00338 - 0.0004394

Introduction

In the realm of mathematics, infinite geometric series hold a special place. These series, characterized by a constant ratio between successive terms, can either converge to a finite sum or diverge to infinity. Understanding the conditions for convergence and the formula for calculating the sum of a convergent series is crucial. In this article, we will delve into the fascinating world of infinite geometric series, exploring their properties and learning how to determine their sums. Our focus will be on a specific example: the series 0.2 - 0.026 + 0.00338 - 0.0004394 + .... We will meticulously analyze this series to ascertain whether it converges, and if so, we will employ the appropriate formula to compute its sum. This journey will not only enhance your understanding of geometric series but also sharpen your analytical skills in the broader field of mathematics. To fully appreciate the concepts, a basic understanding of sequences and series is beneficial, but we will provide a comprehensive explanation to ensure clarity for readers of all levels. Get ready to embark on a mathematical exploration that will unravel the mysteries of infinite geometric series and equip you with the tools to tackle similar problems with confidence.

Understanding Geometric Series

Before we tackle the specific problem at hand, let's solidify our understanding of geometric series. A geometric series is a sequence of numbers where each term is multiplied by a constant factor to obtain the next term. This constant factor is known as the common ratio, often denoted by 'r'. The general form of a geometric series is a + ar + ar^2 + ar^3 + ..., where 'a' is the first term. To illustrate, consider the series 2 + 6 + 18 + 54 + .... Here, the first term 'a' is 2, and the common ratio 'r' is 3 (since each term is multiplied by 3 to get the next term). Geometric series can be finite (having a limited number of terms) or infinite (extending indefinitely). Our primary focus in this article is on infinite geometric series. The behavior of an infinite geometric series, whether it converges or diverges, hinges critically on the value of the common ratio 'r'. Convergence implies that the sum of the infinite terms approaches a finite value, while divergence means the sum grows without bound. A key principle to remember is that an infinite geometric series converges only if the absolute value of the common ratio is less than 1 (|r| < 1). This condition ensures that the terms become progressively smaller, allowing the sum to approach a finite limit. Conversely, if |r| ≥ 1, the series diverges, as the terms either remain constant or increase in magnitude, leading to an unbounded sum. Recognizing this fundamental property is essential for determining whether an infinite geometric series has a meaningful sum. In the next section, we will delve into the formula for calculating the sum of a convergent infinite geometric series, equipping you with the necessary tool to solve problems like the one presented in the title.

Determining Convergence and Finding the Common Ratio

Now that we have a solid grasp of geometric series, let's apply this knowledge to the specific series in question: 0.2 - 0.026 + 0.00338 - 0.0004394 + .... The first crucial step is to determine whether this series is indeed geometric. To do so, we need to check if there's a constant ratio between consecutive terms. We can find this ratio by dividing any term by its preceding term. Let's divide the second term (-0.026) by the first term (0.2): -0.026 / 0.2 = -0.13. Now, let's verify this ratio by dividing the third term (0.00338) by the second term (-0.026): 0.00338 / -0.026 = -0.13. As we can see, the ratio between consecutive terms is consistently -0.13. This confirms that the given series is a geometric series, and the common ratio 'r' is -0.13. Next, we need to determine if this series converges. As discussed earlier, an infinite geometric series converges only if the absolute value of the common ratio is less than 1. In this case, |r| = |-0.13| = 0.13, which is indeed less than 1. Therefore, we can confidently conclude that the given infinite geometric series converges. Since we have established convergence, we can proceed to calculate the sum of the series. The formula for the sum of a convergent infinite geometric series is a / (1 - r), where 'a' is the first term and 'r' is the common ratio. In our case, a = 0.2 and r = -0.13. In the following section, we will apply this formula to find the sum of the series, providing a clear and concise solution to the problem. Understanding the process of determining convergence and finding the common ratio is paramount in dealing with geometric series, and this section has provided a step-by-step guide to achieving this.

Calculating the Sum of the Infinite Geometric Series

With the knowledge that our series converges and having identified the common ratio, we are now ready to calculate the sum of the infinite geometric series 0.2 - 0.026 + 0.00338 - 0.0004394 + .... As we established earlier, the formula for the sum (S) of a convergent infinite geometric series is given by:

S = a / (1 - r)

where 'a' is the first term and 'r' is the common ratio. In our specific case, we have:

a = 0.2 (the first term) r = -0.13 (the common ratio, as determined in the previous section)

Now, we simply substitute these values into the formula:

S = 0.2 / (1 - (-0.13))

Simplifying the expression, we get:

S = 0.2 / (1 + 0.13)

S = 0.2 / 1.13

Performing the division, we find:

S ≈ 0.17699115044

Therefore, the sum of the infinite geometric series 0.2 - 0.026 + 0.00338 - 0.0004394 + ... is approximately 0.17699. This result signifies that as we add more and more terms of the series, the sum gets closer and closer to this value. It's important to note that this is an approximate value due to rounding. However, it provides a very accurate representation of the sum of the infinite series. This calculation demonstrates the power of the formula for the sum of an infinite geometric series, allowing us to find the sum even when dealing with an infinite number of terms. The ability to calculate such sums is a fundamental concept in various areas of mathematics and its applications, including calculus, physics, and engineering. In the concluding section, we will summarize our findings and highlight the key concepts learned in this article.

Conclusion

In this comprehensive exploration, we have successfully navigated the realm of infinite geometric series and determined the sum of the series 0.2 - 0.026 + 0.00338 - 0.0004394 + .... We began by establishing a solid foundation in the understanding of geometric series, defining the common ratio and the conditions for convergence. A key takeaway is that an infinite geometric series converges only if the absolute value of its common ratio is less than 1 (|r| < 1). This condition ensures that the terms of the series diminish sufficiently, allowing the sum to approach a finite value. We then meticulously analyzed the given series, confirming its geometric nature and calculating the common ratio to be -0.13. The absolute value of this ratio, 0.13, is indeed less than 1, which affirmed the convergence of the series. Having established convergence, we employed the formula for the sum of an infinite geometric series, S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. By substituting the values a = 0.2 and r = -0.13 into the formula, we calculated the sum to be approximately 0.17699. This result provides a concrete answer to the problem posed in the title and demonstrates the practical application of the formula. The ability to determine the convergence and calculate the sum of an infinite geometric series is a valuable skill in various mathematical contexts. It highlights the power of mathematical formulas to solve problems involving infinity and provides insights into the behavior of sequences and series. We hope this article has not only provided a solution to the specific problem but also enhanced your understanding of infinite geometric series and their significance in mathematics.