Sum Of Convergent Series ∑[n=1 To ∞] 16/(n(n+2)) Explained

In the realm of mathematical analysis, infinite series hold a position of significant importance. These series, which represent the sum of an infinite number of terms, often exhibit fascinating convergence properties, leading to a finite sum. One such series that piques the interest of mathematicians is the convergent series ∑[n=1 to ∞] 16/(n(n+2)). This article delves into a comprehensive exploration of this series, providing a step-by-step guide to finding its sum. We will unravel the intricacies of this series, employing techniques such as partial fraction decomposition and the concept of telescoping series to arrive at the solution. Understanding these methods will not only equip you to solve this specific problem but also enhance your ability to tackle a wide range of similar challenges in calculus and mathematical analysis.

Before we embark on the journey of finding the sum of the series ∑[n=1 to ∞] 16/(n(n+2)), it is crucial to establish a solid understanding of convergent series. A series is deemed convergent if the sequence of its partial sums approaches a finite limit. In simpler terms, if we keep adding more and more terms of the series, the sum gets closer and closer to a specific value. This value is what we refer to as the sum of the convergent series. On the contrary, a divergent series is one where the sequence of partial sums does not converge to a finite limit; instead, it either grows indefinitely or oscillates without settling on a specific value.

To determine whether a series converges or diverges, mathematicians employ a variety of tests, including the ratio test, the root test, and the integral test. These tests provide conditions under which a series is guaranteed to converge or diverge. For instance, the ratio test states that if the limit of the ratio of consecutive terms is less than 1, the series converges, while if the limit is greater than 1, the series diverges. In the case of the series ∑[n=1 to ∞] 16/(n(n+2)), we can intuitively see that as n becomes very large, the terms of the series approach zero, which hints at the possibility of convergence. However, to rigorously prove convergence and find the sum, we need to employ more sophisticated techniques.

The cornerstone of our approach to finding the sum of the series ∑[n=1 to ∞] 16/(n(n+2)) lies in the technique of partial fraction decomposition. This technique allows us to break down a complex rational expression into simpler fractions, making it easier to analyze and manipulate. In our case, we aim to decompose the fraction 16/(n(n+2)) into the sum of two fractions with simpler denominators. To achieve this, we assume that 16/(n(n+2)) can be expressed in the form A/n + B/(n+2), where A and B are constants that we need to determine.

To find the values of A and B, we multiply both sides of the equation by n(n+2), which clears the denominators. This yields the equation 16 = A(n+2) + Bn. Expanding the right side, we get 16 = An + 2A + Bn. Now, we can group the terms with n and the constant terms, resulting in 16 = (A+B)n + 2A. For this equation to hold true for all values of n, the coefficients of the corresponding terms on both sides must be equal. This gives us two equations: A+B = 0 and 2A = 16. Solving these equations, we find that A = 8 and B = -8. Therefore, we can rewrite the fraction 16/(n(n+2)) as 8/n - 8/(n+2).

With the fraction 16/(n(n+2)) decomposed into simpler fractions, we can now rewrite the original series as ∑[n=1 to ∞] (8/n - 8/(n+2)). This transformation is crucial because it unveils a hidden property of the series: it is a telescoping series. A telescoping series is one where most of the terms cancel out when we write out the partial sums, leaving only a few terms at the beginning and end. This cancellation simplifies the calculation of the sum of the series.

To see the telescoping nature of our series, let's write out the first few terms of the partial sum: (8/1 - 8/3) + (8/2 - 8/4) + (8/3 - 8/5) + (8/4 - 8/6) + .... Notice that the term -8/3 in the first parenthesis cancels out with the term 8/3 in the third parenthesis. Similarly, the term -8/4 in the second parenthesis cancels out with the term 8/4 in the fourth parenthesis. This pattern continues, with most of the terms canceling out. In general, the partial sum S_N, which is the sum of the first N terms of the series, can be written as S_N = 8/1 + 8/2 - 8/(N+1) - 8/(N+2). The beauty of a telescoping series lies in this massive cancellation, which drastically simplifies the expression for the partial sum.

Now that we have an expression for the partial sum S_N, we can find the sum of the infinite series by taking the limit as N approaches infinity. Mathematically, we write this as lim (N→∞) S_N = lim (N→∞) [8/1 + 8/2 - 8/(N+1) - 8/(N+2)]. As N becomes infinitely large, the terms 8/(N+1) and 8/(N+2) approach zero. This is because the denominator becomes infinitely large, making the fraction infinitesimally small. Therefore, the limit simplifies to lim (N→∞) S_N = 8/1 + 8/2 = 8 + 4 = 12.

This result reveals that the sum of the convergent series ∑[n=1 to ∞] 16/(n(n+2)) is 12. This is a remarkable outcome, as it demonstrates that the sum of an infinite number of terms can indeed converge to a finite value. The techniques we employed, namely partial fraction decomposition and the concept of telescoping series, are powerful tools in the arsenal of mathematicians for analyzing and solving problems involving infinite series. The convergence of this series to 12 is not just a numerical result; it's a testament to the elegance and predictability that can be found within the seemingly boundless realm of infinity.

In conclusion, we have successfully navigated the intricacies of the convergent series ∑[n=1 to ∞] 16/(n(n+2)) and determined its sum to be 12. This journey involved a multi-faceted approach, beginning with an understanding of convergent series and the tests used to establish convergence. We then delved into the crucial technique of partial fraction decomposition, which allowed us to simplify the complex rational expression into more manageable components. The transformation of the series into a telescoping series was a pivotal step, enabling the cancellation of terms and the simplification of the partial sum. Finally, by taking the limit of the partial sum as N approached infinity, we arrived at the definitive sum of the series.

The methods employed in this exploration are not confined to this specific example; they represent a broader toolkit applicable to a wide array of problems in calculus and mathematical analysis. Understanding partial fraction decomposition, recognizing telescoping series, and mastering the concept of limits are essential skills for anyone venturing into the world of advanced mathematics. The successful solution of the series ∑[n=1 to ∞] 16/(n(n+2)) serves as a testament to the power and elegance of these techniques. As you continue your mathematical journey, remember that each problem solved is not just an answer obtained, but a step forward in your understanding of the intricate and beautiful patterns that govern the mathematical universe.

This exploration underscores the importance of breaking down complex problems into simpler, more manageable parts. The initial series may have seemed daunting, but by systematically applying appropriate techniques, we were able to unravel its mysteries and arrive at a clear and concise solution. This approach, characterized by methodical analysis and strategic application of mathematical tools, is a cornerstone of problem-solving in mathematics and beyond. As you encounter new challenges, draw upon the lessons learned here, and remember that with the right approach, even the most complex problems can be conquered.