Calculating Series & Decomposing Into Integer & Fractional Parts

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Hey math enthusiasts! Let's dive into a fun problem involving series and some clever calculations. We're going to break down the series, find its value, and then figure out its integer and fractional parts. Sounds good? Let's get started!

Understanding the Series and Our Goal

Alright, first things first, let's get acquainted with the series. We are given:

x = 1/(1*5) + 1/(5*9) + ... + 1/((4n-3)(4n+1))

Our mission, should we choose to accept it (and we definitely do!), is to find the value of x. Once we have x, we'll then calculate its integer part, denoted by [x], and its fractional part, represented by {x}. Remember, the integer part is the whole number part of x, and the fractional part is the decimal part. For example, if x = 3.75, then [x] = 3, and {x} = 0.75. Easy peasy, right? The key here is to find a smart way to simplify the series. We can't just start adding fractions (though we could, eventually!). Instead, we'll try to find a pattern or a trick that makes the calculation much simpler.

Think about this like a detective solving a mystery. We've got a clue – the series – and our goal is to uncover the hidden value of x. This involves some mathematical sleuthing, including breaking down complicated fractions and recognizing patterns. We'll be using techniques that are super useful in calculus and other higher-level math areas. So, understanding this problem sets a strong foundation for future mathematical endeavors. So, grab your pencils and let's get to work on cracking this code! We're not just solving a math problem here; we're building up our problem-solving muscles. That means the tools and tricks we learn today can be applied to solve future problems! The beauty of mathematics lies in its logic and how one concept builds upon the next. This journey isn't just about finding x; it's about appreciating the elegance of the math we will do.

Deconstructing the Fraction

Now, here is the secret sauce: the method of partial fractions. It is a powerful technique that helps us to break down complex fractions into simpler ones. In our series, each term has the form 1/((4n-3)(4n+1)). The goal is to rewrite this fraction as the sum of two other fractions. Imagine that 1/((4n-3)(4n+1)) can be expressed as A/(4n-3) + B/(4n+1). We need to find the values of A and B. To find A and B, we'll clear the denominators by multiplying both sides of the equation by (4n-3)(4n+1). This will leave us with a simpler equation that we can solve. It's like taking a complex jigsaw puzzle and breaking it down into smaller, easier-to-manage pieces. By doing this, we make the series manageable, and pattern recognition becomes a breeze. So, after finding the values of A and B, we can see a beautiful collapse of the sums, which helps us arrive at the value of x in a streamlined manner. This breakdown is not just a trick; it reveals the inner structure of the expression, making it easier to analyze and interpret. So, by employing partial fractions, we transform the initial complex expression into a simpler, more approachable form.

Simplifying the Series

Alright, let's get our hands dirty with some calculations. We aim to rewrite 1/((4n-3)(4n+1)) using partial fractions. After some algebraic manipulation (you'll see in the detailed steps below!), we discover that 1/((4n-3)(4n+1)) = 1/4 * (1/(4n-3) - 1/(4n+1)). This is a crucial breakthrough! Do you see why? Each term in our original series can now be expressed as the difference of two fractions. Now, watch what happens when we start writing out the series using this new form:

x = 1/4 * [(1/1 - 1/5) + (1/5 - 1/9) + (1/9 - 1/13) + ... + (1/(4n-3) - 1/(4n+1))]

Do you see a pattern? Most of the terms cancel each other out! This is known as a telescoping series, and it's a real time-saver. Notice that -1/5 cancels with +1/5, -1/9 cancels with +1/9, and so on. The only terms that remain are the first one, 1/1, and the last one, -1/(4n+1). This means our series simplifies dramatically:

x = 1/4 * [1 - 1/(4n+1)]

By carefully selecting and rearranging terms, we create a situation where most components vanish, simplifying the overall calculation. By doing this simplification, the intricate fractions transform into something more manageable. Remember, the elegance of mathematics lies in these clever simplifications. Always keep an eye out for patterns that let you simplify complicated expressions. The telescoping nature of this series is a perfect example of mathematical beauty, showing how carefully selected terms can beautifully simplify a long sum into a concise form. It's about seeing beyond the surface complexity. By recognizing the pattern and understanding the underlying principles, we can convert a seemingly complex series into something we can easily calculate. This step not only helps us find the value of x but also demonstrates the power of algebraic manipulation in simplifying and solving mathematical problems.

Computing the Value of x

Now that we've simplified the series, calculating x is a piece of cake. We have:

x = 1/4 * [1 - 1/(4n+1)]

To find the value of x, we just need to evaluate this expression. We are not given a specific value for n, but we can analyze what happens as n grows larger. As n becomes very large, the term 1/(4n+1) approaches zero. This is because the denominator (4n+1) grows much faster than the numerator (which is always 1). So, as n goes to infinity, x approaches 1/4. However, we're not given a value for n, so we can assume that n is finite.

x = 1/4 * [1 - 1/(4n+1)] = 1/4 - 1/(16n+4)

We can see that the exact value of x depends on the value of n. For any finite n, x will be slightly less than 1/4. However, without knowing n, we can still find the integer and fractional parts of x by considering the range of possible values.

Determining the Integer and Fractional Parts

Here comes the final step! We need to determine [x] and {x}. We know that x = 1/4 - 1/(16n+4). Since n is a positive integer, the term 1/(16n+4) is always positive. This means that x is always less than 1/4 (or 0.25). So, the integer part of x, [x], will always be 0 because x is less than 1, so [x] must equal 0.

Now, let's find the fractional part, {x}. We know that {x} = x - [x]. Since [x] = 0, then {x} = x - 0 = x. Thus:

{x} = 1/4 - 1/(16n+4)

So, the fractional part of x is 1/4 - 1/(16n+4). For any given value of n, you can plug it into this expression to find the exact value of the fractional part.

Detailed Breakdown

Let's break down the calculations step by step:

  1. Partial Fractions:

    • We start with 1/((4n-3)(4n+1)) and want to express it as A/(4n-3) + B/(4n+1).
    • Multiply both sides by (4n-3)(4n+1) to get 1 = A(4n+1) + B(4n-3).
    • To find A, let 4n-3 = 0, so n = 3/4. Substitute n = 3/4 into the equation, we get 1 = A(4*(3/4)+1) + B*0, so 1 = 4A, and A = 1/4.
    • To find B, let 4n+1 = 0, so n = -1/4. Substitute n = -1/4 into the equation, we get 1 = A*0 + B(4*(-1/4)-3), so 1 = -4B, and B = -1/4.
    • Therefore, 1/((4n-3)(4n+1)) = 1/4 * (1/(4n-3) - 1/(4n+1)).

  2. Rewrite the Series:

    • x = Σ 1/((4n-3)(4n+1)) for n from 1 to some value.
    • Using the partial fraction result:
    • x = Σ 1/4 * (1/(4n-3) - 1/(4n+1))
    • x = 1/4 * Σ (1/(4n-3) - 1/(4n+1))

  3. Telescoping Series:

    • Write out the first few terms of the series:
    • x = 1/4 * [(1/1 - 1/5) + (1/5 - 1/9) + (1/9 - 1/13) + ... + (1/(4n-3) - 1/(4n+1))]
    • Notice how terms cancel each other out.
    • The series simplifies to x = 1/4 * [1 - 1/(4n+1)]

  4. Calculate x:

    • x = 1/4 * [1 - 1/(4n+1)]
    • x = 1/4 - 1/(16n+4)

  5. Determine Integer and Fractional Parts:

    • Since x is less than 1/4, [x] = 0.
    • {x} = x - [x] = x - 0 = x = 1/4 - 1/(16n+4).

In Summary

So, to recap, the key to solving this problem was the use of partial fractions to break down the complex fraction and then the recognition of the telescoping nature of the series. By carefully rearranging the terms, we were able to simplify the series and find the value of x. From there, we easily found the integer and fractional parts of x. This problem beautifully illustrates how powerful algebraic manipulation and pattern recognition can be when tackling mathematical challenges. Keep practicing these techniques, and you'll become a math whiz in no time. Congratulations! You've successfully completed the challenge.

I hope you enjoyed this journey through series and fractions. Keep exploring the wonders of mathematics, and never stop questioning and learning. Happy calculating, everyone!