Power Series Via Partial Fractions: A Step-by-Step Guide

Hey guys! Ever wondered how to break down a complicated fraction into simpler terms and then express it as a power series? Today, we're diving deep into the world of partial fractions and power series to tackle an interesting problem. We'll take a function that looks like a hot mess and transform it into something elegant and useful. So, buckle up, and let's get started!

Understanding the Problem

We are given the function:

f(x) = (x+86) / ((x-5)(x+8))

Our mission, should we choose to accept it (and we do!), is to express this function as a power series, which will look something like:

∑[n=0 to ∞] a_n * x^n

where a_n are the coefficients we need to find. The key to cracking this problem lies in using partial fraction decomposition. This technique allows us to break down the complex rational function into simpler fractions that are easier to manipulate and express as power series. Essentially, we want to rewrite f(x) in the form:

f(x) = A/(x-5) + B/(x+8)

where A and B are constants that we need to determine. Once we have these constants, we can express each term as a power series and combine them to get the power series representation of the original function. The beauty of power series is that they allow us to represent functions as infinite sums of terms involving powers of x, which can be incredibly useful for various mathematical operations and approximations. So, let's roll up our sleeves and dive into the step-by-step process of finding these partial fractions and then converting them into power series.

Step 1: Partial Fraction Decomposition

Our first task is to find the constants A and B such that:

(x+86) / ((x-5)(x+8)) = A/(x-5) + B/(x+8)

To find A and B, we clear the denominators by multiplying both sides by (x-5)(x+8):

x + 86 = A(x+8) + B(x-5)

Now, we can use a couple of clever tricks to solve for A and B. One method is to choose specific values of x that will eliminate one of the variables. Let's start by setting x = 5:

5 + 86 = A(5+8) + B(5-5)

91 = 13A

A = 91/13 = 7

Now, let's set x = -8:

-8 + 86 = A(-8+8) + B(-8-5)

78 = -13B

B = 78/(-13) = -6

So, we have found that A = 7 and B = -6. This means we can rewrite our function as:

f(x) = 7/(x-5) - 6/(x+8)

This decomposition is a crucial step because it breaks down the original complex fraction into two simpler fractions, each of which we can express as a power series using the formula for the sum of an infinite geometric series. By finding these constants, we've essentially unlocked the door to expressing our function in terms of manageable power series components. The next step will involve manipulating these fractions to fit the form of a geometric series, which will allow us to directly write down their power series representations.

Step 2: Expressing as Power Series

Now that we have f(x) = 7/(x-5) - 6/(x+8), we want to express each term as a power series. To do this, we'll manipulate each fraction into the form 1/(1-r), where r is a function of x. This form is perfect because we know that 1/(1-r) = ∑[n=0 to ∞] r^n when |r| < 1. Let's start with the first term:

7/(x-5) = 7/(-5(1 - x/5)) = -7/5 * 1/(1 - x/5)

Now, we can express this as a power series:

-7/5 * ∑[n=0 to ∞] (x/5)^n = ∑[n=0 to ∞] (-7/5^(n+1)) * x^n

This series converges when |x/5| < 1, which means |x| < 5. Next, let's tackle the second term:

-6/(x+8) = -6/(8(1 + x/8)) = -6/8 * 1/(1 + x/8) = -3/4 * 1/(1 - (-x/8))

Expressing this as a power series:

-3/4 * ∑[n=0 to ∞] (-x/8)^n = ∑[n=0 to ∞] (-3/4) * (-1/8)^n * x^n = ∑[n=0 to ∞] (-3 * (-1)^n) / (4 * 8^n) * x^n

This series converges when |-x/8| < 1, which means |x| < 8. So, we have successfully expressed each term in our partial fraction decomposition as a power series. The key here was to manipulate each fraction into the form 1/(1-r), which allowed us to directly apply the formula for the sum of an infinite geometric series. Now, we can combine these two power series to obtain the power series representation of the original function.

Step 3: Combining the Power Series

Now we combine the two power series we found in the previous step:

f(x) = ∑[n=0 to ∞] (-7/5^(n+1)) * x^n + ∑[n=0 to ∞] (-3 * (-1)^n) / (4 * 8^n) * x^n

We can combine these into a single power series by adding the coefficients:

f(x) = ∑[n=0 to ∞] [(-7/5^(n+1)) + (-3 * (-1)^n) / (4 * 8^n)] * x^n

So, the coefficient of x^n is:

a_n = (-7/5^(n+1)) + (-3 * (-1)^n) / (4 * 8^n)

Therefore, the power series representation of the function is:

f(x) = ∑[n=0 to ∞] [(-7/5^(n+1)) + (-3 * (-1)^n) / (4 * 8^n)] * x^n

This series converges for |x| < 5 because that's the smaller of the two radii of convergence for the individual series. This means that the power series representation we found is valid within the interval (-5, 5). Outside this interval, the series will not converge to the original function. So, we have successfully found the power series representation of the given function using partial fractions! We broke down the complex rational function into simpler fractions, expressed each fraction as a power series, and then combined the power series to obtain the final result. This method is a powerful tool for dealing with rational functions and expressing them in a form that is often easier to work with.

Conclusion

Alright, folks! We've successfully navigated the world of partial fractions and power series to find the power series representation of the function f(x) = (x+86) / ((x-5)(x+8)). By using partial fraction decomposition, we simplified the function into manageable terms, expressed each term as a power series using geometric series formulas, and then combined the series to get our final answer. Remember, the key steps were:

1. Partial Fraction Decomposition: Break down the complex fraction into simpler fractions.
2. Expressing as Power Series: Manipulate each fraction into the form 1/(1-r) and use the geometric series formula.
3. Combining the Power Series: Add the coefficients of the individual power series to get the final power series representation.

This technique is super useful in various areas of mathematics, physics, and engineering, so make sure you've got a good grasp of it. Keep practicing, and you'll be a pro in no time! Keep rocking, and see you in the next math adventure!