Finding The Sum Of Series A Step By Step Solution
In the fascinating realm of mathematical series, our goal is to find the sum of this intriguing series:
[ 1/(1 * 2 * 3) + 1/(2 * 3 * 4) + ... + 1/(99 * 100 * 101) ]
This problem, at first glance, may appear daunting, but with the application of clever techniques and a bit of mathematical insight, we can unravel its solution. Let's embark on this mathematical journey together!
Understanding the Series
Before diving into calculations, it's crucial to understand the series we're dealing with. This series is a finite sum of terms, each of which has a specific form. The general term of the series can be expressed as:
1/[n * (n + 1) * (n + 2)]
where n ranges from 1 to 99. This form suggests that we might be able to use a method called partial fraction decomposition to simplify each term and reveal a pattern that allows us to calculate the sum.
Partial fraction decomposition is a powerful technique that allows us to break down complex fractions into simpler ones. In this case, we aim to express the general term as a sum of simpler fractions, each with a denominator that is a factor of the original denominator. This will allow us to rewrite the series in a way that makes it easier to sum.
The Power of Partial Fraction Decomposition
Our first strategic move is to employ partial fraction decomposition. This technique allows us to break down complex fractions into simpler, more manageable components. For the general term of our series, 1/[n(n+1)(n+2)], we seek to express it in the following form:
1/[n(n+1)(n+2)] = A/n + B/(n+1) + C/(n+2)
Here, A, B, and C are constants that we need to determine. To find these constants, we multiply both sides of the equation by n(n+1)(n+2), which gives us:
1 = A(n+1)(n+2) + Bn(n+2) + Cn(n+1)
Now, we can solve for A, B, and C by strategically choosing values for n.
- Let's set n = 0. This simplifies the equation to 1 = A(1)(2), so A = 1/2.
- Next, let n = -1. The equation becomes 1 = B(-1)(1), so B = -1.
- Finally, let n = -2. We get 1 = C(-2)(-1), which gives us C = 1/2.
With the values of A, B, and C in hand, we can rewrite the general term as:
1/[n(n+1)(n+2)] = (1/2)/n - 1/(n+1) + (1/2)/(n+2)
This decomposition is the key to unlocking the sum of the series. By expressing each term in this way, we create a telescoping series, where many terms cancel each other out, leaving us with a much simpler expression to evaluate.
Unveiling the Telescoping Nature
Now comes the crucial step of recognizing the telescoping nature of the series after partial fraction decomposition. When we substitute the decomposition back into the original series, we get:
[ (1/2)/1 - 1/2 + (1/2)/3 ] + [ (1/2)/2 - 1/3 + (1/2)/4 ] + [ (1/2)/3 - 1/4 + (1/2)/5 ] + ... + [ (1/2)/99 - 1/100 + (1/2)/101 ]
Notice the beautiful pattern of cancellation that emerges. The -1/2 in the first group cancels with the (1/2)/2 in the second group. Similarly, the -1/3 in the second group cancels with the (1/2)/3 in the first group, and so on. This cascading cancellation is the hallmark of a telescoping series.
When we carefully examine the series, we see that most of the terms vanish due to cancellation. The terms that survive are those at the beginning and the end of the series. Specifically, we are left with:
(1/2)/1 - 1/2 + (1/2)/2 + (1/2)/100 - 1/100 + (1/2)/101
This significantly simplifies the calculation. We've transformed a seemingly complex series into a simple sum of a few terms.
Calculating the Sum with Precision
With the telescoping effect having worked its magic, we are now in a position to calculate the sum with precision. We are left with the following terms after the cancellations:
(1/2)/1 - 1/2 + 1/4 + (1/2)/101
Let's simplify this expression step by step:
- First, we combine the fractions with a common denominator:
1/2 - 1/2 + 1/4 + 1/202 = 1/4 + 1/202
- Next, we find a common denominator for 1/4 and 1/202, which is 404:
(101/404) + (2/404) = 103/404
Therefore, the sum of the series is 103/404. This is the precise answer we've been seeking. The journey from the initial series to this final result highlights the power of mathematical techniques like partial fraction decomposition and the elegance of telescoping series.
Generalizing the Approach for Similar Series
Having conquered this particular series, it's valuable to generalize the approach so we can tackle similar problems in the future. The key techniques we employed – partial fraction decomposition and recognizing the telescoping pattern – are applicable to a wide range of series problems.
Partial Fraction Decomposition: This technique is particularly useful when dealing with series where the general term is a rational function (a fraction where the numerator and denominator are polynomials). The goal is to break down the complex fraction into simpler fractions with denominators that are factors of the original denominator. This often reveals hidden patterns and simplifies the summation process.
Telescoping Series: Keep an eye out for series where terms cancel each other out in a cascading fashion. This often happens when the general term can be expressed as a difference of two terms, such as f(n) - f(n+1). Recognizing this pattern can dramatically simplify the calculation of the sum.
By mastering these techniques, you'll be well-equipped to tackle a variety of series problems. Remember, practice is key. The more you work with these concepts, the more intuitive they will become.
Conclusion The Beauty of Mathematical Problem-Solving
In conclusion, we have successfully navigated the problem of finding the sum of the series:
[ 1/(1 * 2 * 3) + 1/(2 * 3 * 4) + ... + 1/(99 * 100 * 101) ]
by employing the powerful techniques of partial fraction decomposition and recognizing the telescoping nature of the resulting series. The final sum, 103/404, is a testament to the elegance and precision of mathematics.
This journey underscores the beauty of mathematical problem-solving. By breaking down a complex problem into smaller, more manageable steps, we can uncover hidden patterns and arrive at elegant solutions. The techniques we've explored here are not only valuable for solving specific problems but also for developing a deeper understanding of mathematical principles.
So, embrace the challenge of mathematical exploration, and may your journey be filled with insightful discoveries!
Sum of series, Partial fraction decomposition, Telescoping series, Mathematical series, Series summation, Fraction decomposition, Mathematical problem solving, Series calculation, General term, Finite sum
