Calculating The Sum Of The Series S = 1/(36) + 1/(69) + 1/(912) + ... + 1/(3033)

Jul 15, 2025 by ADMIN 85 views

Unveiling the Sum of the Series: S = 1/(3*6) + 1/(6*9) + 1/(9*12) + ... + 1/(30*33)

Introduction to the Series and Its Significance

In the realm of mathematics, series play a pivotal role in understanding patterns, sequences, and the behavior of numbers. This article delves into the intricacies of a specific series: S = 1/(36) + 1/(69) + 1/(912) + ... + 1/(3033). This series, at first glance, might seem like a simple addition of fractions, but a closer examination reveals a fascinating pattern that allows us to calculate its sum efficiently. The series belongs to a broader class of mathematical expressions known as telescoping series, where intermediate terms cancel out, leading to a simplified final result. Understanding such series is crucial in various fields, including calculus, where they appear in the context of infinite sums and convergence. Furthermore, the techniques used to solve this specific series can be applied to a wide range of similar problems, making it a valuable exercise in mathematical problem-solving. This exploration will not only provide the solution to the given series but also shed light on the underlying principles and methodologies that are applicable in more complex mathematical scenarios. The ability to recognize patterns and apply appropriate techniques is a hallmark of mathematical proficiency, and this article aims to enhance that ability in the reader. The process of deconstructing the series, identifying the pattern, and applying the telescoping technique will be thoroughly explained, ensuring a clear understanding for readers with varying levels of mathematical background. By the end of this article, readers will not only know the sum of the series but also appreciate the beauty and elegance of mathematical problem-solving.

Deconstructing the Series: Identifying the Pattern

To effectively compute the sum of the series S = 1/(36) + 1/(69) + 1/(912) + ... + 1/(3033), the first crucial step is to recognize the underlying pattern within the terms. Observing the denominators, we see a sequence of products where each term is formed by multiplying two numbers that are multiples of 3. Specifically, the denominators can be expressed as 36, 69, 912, and so on, up to 3033. This indicates a consistent structure where each subsequent term's factors increase by 3. The key to solving this series lies in expressing each fraction as a difference of two simpler fractions. This technique, known as partial fraction decomposition, is particularly useful for telescoping series. Notice that each term in the series can be written in the form 1/((3n)(3n+3)), where 'n' is an integer that starts from 1 and increases until 10. This general form allows us to represent the entire series in a compact manner and facilitates the application of partial fraction decomposition. The goal is to rewrite each fraction 1/((3n)(3n+3)) as A/(3n) + B/(3n+3), where A and B are constants. Finding the appropriate values for A and B is essential for the telescoping effect to occur. This decomposition will allow us to cancel out intermediate terms when the series is summed, leading to a simplified expression. The process of finding A and B involves algebraic manipulation and solving a system of equations. Once the decomposition is achieved, the series will transform into a form where the cancellations become evident, making the summation straightforward. This meticulous deconstruction of the series is paramount to revealing its underlying structure and paving the way for an elegant solution. The ability to identify such patterns and apply appropriate decomposition techniques is a fundamental skill in mathematical analysis.

Applying Partial Fraction Decomposition

Having identified the pattern in the series S = 1/(36) + 1/(69) + 1/(912) + ... + 1/(3033), the next step is to apply partial fraction decomposition. As discussed earlier, each term in the series can be represented as 1/((3n)(3n+3)). Our aim is to express this fraction as a difference of two simpler fractions: A/(3n) + B/(3n+3). To find the values of A and B, we set up the equation:

1/((3n)(3n+3)) = A/(3n) + B/(3n+3)

Multiplying both sides by (3n)(3n+3) to clear the denominators, we get:

1 = A(3n+3) + B(3n)

This equation must hold true for all values of 'n'. To solve for A and B, we can choose specific values of 'n' that simplify the equation. Let's first set n = 0:

1 = A(3(0)+3) + B(3(0))

1 = 3A

A = 1/3

Now, to find B, we can choose another convenient value for 'n'. Let's set n = -1:

1 = A(3(-1)+3) + B(3(-1))

1 = 0 - 3B

B = -1/3

Thus, we have found that A = 1/3 and B = -1/3. This means that each term in the series can be decomposed as follows:

1/((3n)(3n+3)) = (1/3)/(3n) - (1/3)/(3n+3)

This decomposition is the key to unlocking the telescoping nature of the series. By rewriting each term in this form, we create a structure where intermediate terms will cancel out when the series is summed. The factor of 1/3 can be factored out of the entire series, simplifying the summation process. The decomposed form clearly shows the difference between two fractions, setting the stage for the telescoping effect. This application of partial fraction decomposition is a powerful technique in simplifying complex fractions and is widely used in calculus and other areas of mathematics. The careful and methodical approach to finding A and B is crucial for the correct decomposition and subsequent solution of the series.

The Telescoping Effect: Cancelling Intermediate Terms

With each term of the series S = 1/(36) + 1/(69) + 1/(912) + ... + 1/(3033) decomposed into partial fractions, we can now observe the telescoping effect. Recall that we expressed each term as:

1/((3n)(3n+3)) = (1/3)/(3n) - (1/3)/(3n+3)

Substituting n = 1, 2, 3, ..., 10 into this expression, we get the following terms:

n = 1: (1/3)/(31) - (1/3)/(32) = (1/3)(1/3 - 1/6)

n = 2: (1/3)/(32) - (1/3)/(33) = (1/3)(1/6 - 1/9)

n = 3: (1/3)/(33) - (1/3)/(34) = (1/3)(1/9 - 1/12)

...

n = 10: (1/3)/(310) - (1/3)/(311) = (1/3)(1/30 - 1/33)

When we sum these terms, we notice a remarkable pattern: the negative term in each expression cancels out with the positive term in the subsequent expression. This is the essence of the telescoping series. Specifically, the -1/6 in the first term cancels with the +1/6 in the second term, the -1/9 in the second term cancels with the +1/9 in the third term, and so on. This cancellation continues until we reach the last term. The only terms that survive the cancellation are the first positive term (1/3)(1/3) and the last negative term (1/3)(-1/33). Therefore, the sum of the series simplifies to:

S = (1/3)[(1/3) - (1/33)]

This telescoping effect drastically reduces the complexity of the summation. Instead of adding ten fractions individually, we only need to consider the first and last terms. This elegant cancellation is a hallmark of telescoping series and highlights the power of partial fraction decomposition in simplifying mathematical expressions. The ability to recognize and exploit this telescoping behavior is a valuable skill in various areas of mathematics, particularly in calculus and analysis. The next step is to simplify the expression and arrive at the final numerical value for the sum of the series.

Calculating the Final Sum

Having observed the telescoping effect and simplified the series S = 1/(36) + 1/(69) + 1/(912) + ... + 1/(3033) to S = (1/3)[(1/3) - (1/33)], we can now proceed with the final calculation to determine the sum. The expression within the brackets involves subtracting two fractions, which requires finding a common denominator. The least common multiple of 3 and 33 is 33. Therefore, we rewrite the expression as:

S = (1/3)[(11/33) - (1/33)]

Now, we can subtract the fractions:

S = (1/3)(10/33)

Finally, we multiply the fractions:

S = 10/99

Thus, the sum of the series S is 10/99. This result demonstrates the power of the telescoping technique in simplifying complex summations. The initial series, with its ten terms, might have seemed daunting to calculate directly. However, by recognizing the pattern, applying partial fraction decomposition, and exploiting the telescoping effect, we arrived at the solution with relative ease. This process highlights the importance of problem-solving strategies in mathematics. Identifying the appropriate technique for a given problem can significantly reduce the computational effort and lead to an elegant solution. The value 10/99 is a precise representation of the sum, and it provides a concrete answer to the initial question. The journey from the initial series to the final sum showcases the beauty and efficiency of mathematical methods. The ability to break down a complex problem into smaller, manageable steps is a key skill in mathematics and other disciplines. This example serves as a valuable illustration of how such skills can be applied to achieve a clear and concise solution.

Conclusion: The Elegance of Telescoping Series

In conclusion, the sum of the series S = 1/(36) + 1/(69) + 1/(912) + ... + 1/(3033) is 10/99. This result was obtained by carefully deconstructing the series, recognizing the underlying pattern, applying partial fraction decomposition, and exploiting the telescoping effect. The telescoping series is a beautiful example of how mathematical structures can lead to elegant solutions. The key to solving this series was recognizing that each term could be expressed as a difference of two fractions, leading to the cancellation of intermediate terms. This technique is not only applicable to this specific series but also to a broader class of similar problems in mathematics. The process of partial fraction decomposition, where a complex fraction is broken down into simpler fractions, is a valuable tool in calculus, analysis, and other areas of mathematics. The ability to identify patterns, apply appropriate techniques, and simplify expressions is crucial for mathematical problem-solving. This article has demonstrated a methodical approach to solving a series, starting from the initial observation of the pattern to the final calculation of the sum. The steps involved, including partial fraction decomposition and the recognition of the telescoping effect, are fundamental concepts in mathematical analysis. The solution to this series not only provides a numerical answer but also illustrates the power and elegance of mathematical methods. The journey from the initial series to the final sum highlights the importance of problem-solving strategies and the beauty of mathematical structures. Understanding telescoping series and related techniques enhances one's mathematical proficiency and provides a valuable foundation for tackling more complex problems. The process of solving this series serves as a testament to the interconnectedness of mathematical concepts and the importance of a systematic approach to problem-solving.