Evaluating The Series B = 3/(1 X 3) + 3/(3 X 5) + 3/(5 X 7) + ... + 3/(99 X 101)

In this article, we will delve into the evaluation of the series B, which is defined as the sum of fractions with a specific pattern in their denominators. Specifically, the series is given by: B = 3/(1 x 3) + 3/(3 x 5) + 3/(5 x 7) + ... + 3/(99 x 101). Series evaluation is a fundamental concept in mathematics, often encountered in calculus and analysis. This particular series showcases a pattern that allows us to use a technique called partial fraction decomposition to simplify and ultimately find the sum. Understanding how to evaluate such series is crucial for anyone studying mathematical analysis or related fields. This problem not only tests our ability to apply mathematical formulas but also our pattern recognition skills. The series' structure, with the denominators being products of consecutive odd numbers, is key to unlocking the solution. The goal is to break down each fraction into simpler components, allowing for a telescoping effect where intermediate terms cancel out, leaving us with a manageable expression to calculate the final sum.

To effectively evaluate the series, it's crucial to first understand the underlying structure and pattern. The series B consists of terms where each fraction has a numerator of 3, and the denominator is the product of two consecutive odd numbers. This pattern is the key to finding a simplified method for summation. Recognizing this pattern allows us to consider techniques like partial fraction decomposition, which is often used to simplify such expressions. Partial fraction decomposition breaks down a complex fraction into simpler fractions with denominators that are factors of the original denominator. In this case, each term can be decomposed into the difference of two fractions, making the summation process significantly easier. The general form of the terms in the series is 3/((2n-1)(2n+1)), where n ranges from 1 to 50. Understanding this general form helps us to apply the partial fraction decomposition method systematically across all terms in the series. By recognizing and utilizing the pattern, we can transform the series into a form where most of the terms cancel out, a phenomenon known as telescoping, leading to a straightforward calculation of the sum.

The method of partial fraction decomposition is a powerful technique for simplifying rational expressions, particularly useful when dealing with series like the one presented. This method allows us to break down a complex fraction into simpler fractions, making it easier to find a pattern and sum the series. For the series B, each term is of the form 3/((2n-1)(2n+1)). The goal is to express this fraction as the difference of two simpler fractions. Partial fraction decomposition involves expressing the given fraction as a sum or difference of fractions with simpler denominators. In our case, we want to find constants A and B such that 3/((2n-1)(2n+1)) = A/(2n-1) + B/(2n+1). By solving for A and B, we can rewrite each term in the series in a more manageable form. The process typically involves multiplying both sides of the equation by the common denominator ((2n-1)(2n+1)) and then equating the coefficients of like terms. This results in a system of equations that can be solved to find the values of A and B. Once we have decomposed the fractions, we can substitute them back into the series and look for terms that cancel each other out, leading to a simplified expression for the sum. This method is a cornerstone of many calculus and analysis problems involving series and sequences.

Let's go through a step-by-step solution to evaluate the series B using partial fraction decomposition. This will involve breaking down each term into simpler fractions and then summing the series to find the result. We begin by expressing the general term of the series, 3/((2n-1)(2n+1)), as the sum of two simpler fractions: A/(2n-1) + B/(2n+1). The first step is to find the constants A and B. To do this, we multiply both sides of the equation by the common denominator (2n-1)(2n+1), which gives us: 3 = A(2n+1) + B(2n-1). Next, we expand and group the terms: 3 = (2A + 2B)n + (A - B). For this equation to hold true for all n, the coefficients of n must be equal to zero, and the constant terms must be equal. This gives us two equations: 2A + 2B = 0 and A - B = 3. Solving this system of equations, we find that A = 3/2 and B = -3/2. Now we can rewrite the general term as: 3/((2n-1)(2n+1)) = (3/2)/(2n-1) - (3/2)/(2n+1). Substituting these back into the series, we get a telescoping series where most terms cancel out. This simplification allows us to find the sum by considering only the first and last terms.

Now, let's apply partial fraction decomposition to the entire series. We've established that each term 3/((2n-1)(2n+1)) can be written as (3/2)/(2n-1) - (3/2)/(2n+1). So, the series B can be rewritten as: B = (3/2)/(1) - (3/2)/(3) + (3/2)/(3) - (3/2)/(5) + (3/2)/(5) - (3/2)/(7) + ... + (3/2)/(99) - (3/2)/(101). Notice how most of the terms cancel each other out. This is the telescoping effect, a key characteristic of series that can be simplified using partial fraction decomposition. The terms -(3/2)/(3) and +(3/2)/(3) cancel, as do -(3/2)/(5) and +(3/2)/(5), and so on. This pattern continues throughout the series, leaving only the first and last terms. After cancellation, we are left with: B = (3/2)/(1) - (3/2)/(101). Now, we can simplify this expression to find the sum of the series. This process demonstrates the power of partial fraction decomposition in turning a seemingly complex series into a manageable calculation. By breaking down each term and recognizing the telescoping pattern, we can efficiently find the sum.

After applying partial fraction decomposition and observing the telescoping effect, we are left with a simplified expression for the series B. We have: B = (3/2)/(1) - (3/2)/(101). To find the sum, we need to perform the subtraction. First, we can factor out the common factor of 3/2, giving us: B = (3/2) * (1 - 1/101). Next, we find a common denominator to subtract the fractions inside the parentheses. The common denominator is 101, so we rewrite 1 as 101/101. This gives us: B = (3/2) * (101/101 - 1/101). Now, we can subtract the fractions: B = (3/2) * (100/101). Finally, we multiply the fractions: B = (3 * 100) / (2 * 101) = 300 / 202. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2: B = 150 / 101. Therefore, the sum of the series B is 150/101. This calculation demonstrates how partial fraction decomposition, combined with the telescoping effect, can greatly simplify the process of summing series.

After all the steps, we've successfully evaluated the series B. We used partial fraction decomposition to simplify the terms, recognized the telescoping effect, and performed the necessary calculations to arrive at the final result. The sum of the series B = 3/(1 x 3) + 3/(3 x 5) + 3/(5 x 7) + ... + 3/(99 x 101) is 150/101. This result highlights the elegance and efficiency of partial fraction decomposition in handling series with specific patterns. By breaking down the complex fractions into simpler components, we were able to transform the series into a form where most of the terms canceled out, leaving us with a straightforward calculation. The final answer, 150/101, is a precise value that represents the sum of the given series. This exercise demonstrates the importance of pattern recognition and the application of appropriate mathematical techniques in solving problems involving series and sequences. Understanding these methods is crucial for further studies in calculus, analysis, and related fields.

In conclusion, we have successfully evaluated the series B = 3/(1 x 3) + 3/(3 x 5) + 3/(5 x 7) + ... + 3/(99 x 101) using the technique of partial fraction decomposition. This method allowed us to rewrite the complex fractions as simpler terms, leading to a telescoping series where most terms canceled out. This simplification made it possible to calculate the sum of the series efficiently. The final result, 150/101, represents the exact value of the sum. This problem illustrates the power of mathematical techniques in simplifying complex expressions and finding solutions. Partial fraction decomposition is a valuable tool in the repertoire of any mathematician or student studying calculus and analysis. The ability to recognize patterns and apply appropriate methods is crucial for problem-solving in these fields. The process of evaluating this series not only provides a specific answer but also reinforces the understanding of fundamental mathematical concepts. By mastering these techniques, one can approach a wide range of problems involving series and sequences with confidence.