Solving The Infinite Series Σ(2r-1)(9/11)^r A Step-by-Step Guide

Introduction: Delving into the Realm of Infinite Series

In the captivating world of mathematics, infinite series hold a special allure. These seemingly endless sums often converge to finite values, revealing hidden patterns and elegant relationships. One such intriguing series is the expression Σ(2r-1)(9/11)^r, where r ranges from 1 to infinity. This series, while appearing complex at first glance, can be deciphered using fundamental mathematical principles and techniques. In this comprehensive exploration, we will embark on a journey to unravel the intricacies of this series, understand its convergence, and ultimately determine its precise value. Our main keyword here is the infinite series, and we will delve deep into understanding the mathematical principles behind it. This exploration will not only provide the solution to the given problem but also enhance your understanding of series and sequences in general. This foundational knowledge is crucial for various advanced mathematical concepts and applications in fields like physics and engineering. The journey starts with understanding the nature of the series itself, recognizing its components, and then applying appropriate methods to simplify and solve it. This process will involve breaking down the series into manageable parts and using established theorems and techniques to evaluate the sum. The goal is to not only arrive at the correct answer but also to appreciate the elegance and power of mathematical tools in solving complex problems. Throughout this article, we will emphasize clarity and step-by-step explanations to ensure that the concepts are accessible and understandable. Whether you are a student preparing for an exam or simply a mathematics enthusiast, this guide aims to provide a thorough and insightful understanding of the infinite series in question.

Dissecting the Series: Identifying Patterns and Components

To effectively tackle the series Σ(2r-1)(9/11)^r, it is crucial to first dissect its structure and identify the underlying patterns. The series involves a sum of terms, each of which is a product of two components: (2r-1) and (9/11)^r. The term (2r-1) represents a sequence of odd numbers (1, 3, 5, 7, ...), while (9/11)^r represents a geometric sequence with a common ratio of 9/11. The interplay between these two sequences determines the behavior of the overall series. Understanding this interplay is key to finding the sum. The series can be seen as a combination of an arithmetic progression (2r-1) and a geometric progression (9/11)^r. The arithmetic progression contributes a linearly increasing factor, while the geometric progression contributes an exponentially decreasing factor. The balance between these two factors dictates whether the series converges and, if so, what its sum will be. To fully grasp this dynamic, we can expand the series for a few terms: (2(1)-1)(9/11)^1 + (2(2)-1)(9/11)^2 + (2(3)-1)(9/11)^3 + .... This expansion helps visualize how each term contributes to the overall sum and provides insights into the convergence behavior. It's also important to recognize that the common ratio (9/11) of the geometric part is less than 1, which is a crucial indicator of convergence. A geometric series converges if the absolute value of the common ratio is less than 1. However, the presence of the (2r-1) term complicates matters, as it prevents us from directly applying the formula for the sum of a simple geometric series. Therefore, we need to employ a more sophisticated approach to find the sum. This approach will likely involve manipulating the series to separate the arithmetic and geometric components and then using differentiation or other techniques to arrive at a closed-form expression for the sum. The process of dissecting the series and understanding its components is a fundamental step in solving any mathematical problem, especially those involving infinite sums.

The Power of Differentiation: A Technique for Summation

One of the most effective techniques for evaluating series that combine arithmetic and geometric progressions is the method of differentiation. This technique leverages the power of calculus to transform the series into a more manageable form. The core idea is to start with a known geometric series and then differentiate it term by term to introduce the desired arithmetic component. Let's consider the geometric series: S(x) = Σ x^r, where r ranges from 1 to infinity and |x| < 1. We know that the sum of this series is given by S(x) = x / (1 - x). Now, if we differentiate both sides of this equation with respect to x, we obtain: S'(x) = Σ r * x^(r-1) = 1 / (1 - x)^2. This differentiation step is crucial because it introduces the 'r' term, which is characteristic of an arithmetic progression. By multiplying both sides by x, we get: x * S'(x) = Σ r * x^r = x / (1 - x)^2. Notice that we now have a series where each term includes 'r' multiplied by a power of x. This is a significant step towards matching the structure of our target series, Σ(2r-1)(9/11)^r. The next step involves manipulating this derived series to incorporate the (2r-1) term. We can rewrite (2r-1) as 2r - 1 and then express our target series as the difference of two series: 2 * Σ r * (9/11)^r - Σ (9/11)^r. We already have an expression for Σ r * x^r, so we can substitute x = 9/11 into that expression. We also know the sum of the simple geometric series Σ (9/11)^r, which we can calculate directly. By combining these results, we can find the sum of the original series. The method of differentiation highlights the interconnectedness of different areas of mathematics, particularly calculus and series. It demonstrates how differentiation can be used not only to find derivatives but also to solve summation problems. This technique is a powerful tool in the mathematician's arsenal and is applicable to a wide range of series problems.

Applying the Technique: A Step-by-Step Solution

Now, let's apply the differentiation technique to find the sum of our series, Σ(2r-1)(9/11)^r. Following the steps outlined in the previous section, we start with the geometric series S(x) = Σ x^r = x / (1 - x), for |x| < 1. Differentiating both sides with respect to x, we get S'(x) = Σ r * x^(r-1) = 1 / (1 - x)^2. Multiplying by x, we have x * S'(x) = Σ r * x^r = x / (1 - x)^2. Now, we want to find the sum of Σ(2r-1)(9/11)^r. We can rewrite this as 2 * Σ r * (9/11)^r - Σ (9/11)^r. Let's first calculate Σ r * (9/11)^r. Substituting x = 9/11 into the expression x / (1 - x)^2, we get: Σ r * (9/11)^r = (9/11) / (1 - 9/11)^2 = (9/11) / (2/11)^2 = (9/11) / (4/121) = (9/11) * (121/4) = 99/4. Next, we calculate the sum of the geometric series Σ (9/11)^r. Using 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, we have: Σ (9/11)^r = (9/11) / (1 - 9/11) = (9/11) / (2/11) = 9/2. Now, we can substitute these values back into our expression: 2 * Σ r * (9/11)^r - Σ (9/11)^r = 2 * (99/4) - (9/2) = 99/2 - 9/2 = 90/2 = 45. Therefore, the sum of the series Σ(2r-1)(9/11)^r is 45. This step-by-step solution demonstrates the power and efficiency of the differentiation technique. By carefully applying the steps and using the known formulas for geometric series and their derivatives, we were able to arrive at the solution in a systematic and clear manner. This approach not only provides the answer but also reinforces the understanding of the underlying mathematical principles.

Evaluating the Options: Identifying the Correct Answer

Having calculated the sum of the series Σ(2r-1)(9/11)^r to be 45, we now turn our attention to evaluating the given options to identify the correct answer. The options are: (A) 45, (B) 55, (C) sum of first nine natural numbers, and (D) sum of first ten natural numbers. Comparing our calculated sum of 45 with the options, we can immediately see that option (A) is the correct answer. To further confirm our result, let's examine the other options. Option (B) is 55, which is clearly not equal to 45. Option (C) refers to the sum of the first nine natural numbers. The sum of the first n natural numbers is given by the formula n(n+1)/2. For n = 9, this sum is 9(10)/2 = 45. So, option (C) is also equal to 45. Option (D) refers to the sum of the first ten natural numbers. For n = 10, this sum is 10(11)/2 = 55, which is not equal to 45. Therefore, both options (A) and (C) are correct, as the sum of the series is indeed 45, and the sum of the first nine natural numbers is also 45. However, since option (A) directly states the value of the sum, it is the most straightforward and appropriate answer. This process of evaluating the options highlights the importance of not only finding the solution but also verifying its correctness by comparing it with the given choices. It also demonstrates that sometimes, there can be multiple ways to express the same answer, as seen in this case with options (A) and (C). The ability to recognize and relate different mathematical expressions is a crucial skill in problem-solving.

Conclusion: Reflecting on the Journey and the Power of Mathematical Tools

In this comprehensive exploration, we have successfully navigated the intricacies of the infinite series Σ(2r-1)(9/11)^r and determined its sum to be 45. Our journey began with dissecting the series and identifying its components, recognizing the interplay between the arithmetic and geometric progressions. We then employed the powerful technique of differentiation to transform the series into a more manageable form, leveraging the relationship between geometric series and their derivatives. By carefully applying the steps and utilizing established formulas, we arrived at the solution in a systematic and clear manner. Finally, we evaluated the given options and confirmed that our calculated sum matched option (A), and also recognized that option (C) represented the same value. This exploration underscores the elegance and power of mathematical tools in solving complex problems. The technique of differentiation, in particular, demonstrates the interconnectedness of different areas of mathematics and its ability to transform seemingly intractable problems into solvable ones. The journey through this problem also highlights the importance of a structured and methodical approach to problem-solving. By breaking down the problem into smaller, manageable steps and applying appropriate techniques, we were able to navigate the complexities and arrive at the correct solution. Furthermore, this exploration serves as a valuable learning experience, reinforcing the understanding of series and sequences, geometric progressions, differentiation, and the application of these concepts in problem-solving. The skills and insights gained from this exercise will undoubtedly be beneficial in tackling other mathematical challenges. In conclusion, the infinite series Σ(2r-1)(9/11)^r stands as a testament to the beauty and power of mathematics, and our journey to unravel its sum has been a rewarding and enlightening experience.