Power Series Representation Of F(x) = 7/(4-x)

by Andrew McMorgan 46 views

Hey math enthusiasts! Today, we're diving into the fascinating world of power series and how to represent a function using them. Specifically, we'll tackle the function f(x) = 7/(4-x). Our goal is to find its power series representation and, importantly, determine the interval of convergence for this series. So, grab your thinking caps, and let's get started!

Understanding Power Series

Before we jump into the specifics of our function, let's quickly recap what a power series actually is. A power series is essentially an infinite series of the form:

∑[n=0 to ∞] cₙ(x - a)ⁿ = c₀ + c₁(x - a) + c₂(x - a)² + c₃(x - a)³ + ...

Where:

  • cₙ are the coefficients of the series.
  • x is the variable.
  • a is the center of the series.

The key idea is that we're trying to express a function as an infinite sum of powers of (x - a). This representation can be incredibly useful for various mathematical operations, like differentiation and integration, which are often easier to perform on power series than on the original functions.

Now, a crucial aspect of any power series is its interval of convergence. Not all values of x will make the series converge (i.e., have a finite sum). The interval of convergence tells us the range of x values for which the series does converge. Typically, this interval is centered around 'a' and has a certain radius of convergence.

Why is understanding power series so important, you ask? Well, they're used extensively in various fields, from solving differential equations to approximating complex functions. They're a fundamental tool in the mathematician's toolkit, and mastering them opens up a world of possibilities.

Finding the Power Series Representation

Okay, let's get back to our function, f(x) = 7/(4-x). Our mission is to find a power series that represents this function. The trick here is to manipulate the function into a form that resembles the sum of a geometric series. Remember the formula for an infinite geometric series:

∑[n=0 to ∞] arⁿ = a / (1 - r), where |r| < 1

This formula is our golden ticket! We need to rewrite f(x) so that it looks like a / (1 - r). Here's how we do it:

  1. Factor out a 4 from the denominator: f(x) = 7 / (4(1 - x/4))

  2. Rewrite as a product: f(x) = (7/4) * [1 / (1 - x/4)]

Now, we have something that looks very similar to our geometric series formula! We can see that:

  • a = 7/4
  • r = x/4

Therefore, we can express f(x) as a power series:

f(x) = (7/4) * ∑[n=0 to ∞] (x/4)ⁿ

  1. Simplify the expression: f(x) = ∑[n=0 to ∞] (7/4) * (xⁿ / 4ⁿ) f(x) = ∑[n=0 to ∞] (7xⁿ) / (4ⁿ⁺¹)

And there you have it! We've found a power series representation for f(x) = 7/(4-x). It's the infinite sum ∑[n=0 to ∞] (7xⁿ) / (4ⁿ⁺¹). This series represents our function, but we're not done yet. We still need to determine the interval of convergence.

Determining the Interval of Convergence

The interval of convergence is crucial because it tells us for what values of x the power series actually converges to f(x). To find it, we'll use the ratio test. The ratio test is a powerful tool for determining the convergence of a series. It states that if:

lim [n→∞] |aₙ₊₁ / aₙ| < 1, the series converges. lim [n→∞] |aₙ₊₁ / aₙ| > 1, the series diverges. lim [n→∞] |aₙ₊₁ / aₙ| = 1, the test is inconclusive.

Where aₙ represents the nth term of our series. In our case, aₙ = (7xⁿ) / (4ⁿ⁺¹).

Let's apply the ratio test:

  1. Find aₙ₊₁: aₙ₊₁ = (7xⁿ⁺¹) / (4⁽ⁿ⁺¹⁾⁺¹) aₙ₊₁ = (7xⁿ⁺¹) / (4ⁿ⁺²)

  2. Calculate |aₙ₊₁ / aₙ|: |(7xⁿ⁺¹) / (4ⁿ⁺²) / (7xⁿ) / (4ⁿ⁺¹)| |(7xⁿ⁺¹) / (4ⁿ⁺²) * (4ⁿ⁺¹) / (7xⁿ)|

  3. Simplify the expression: |x / 4|

  4. Take the limit as n approaches infinity: lim [n→∞] |x / 4| = |x / 4|

For the series to converge, we need |x / 4| < 1. This inequality gives us:

  • |x| < 4
  • -4 < x < 4

This tells us that the interval of convergence is centered at 0 (our center 'a') and has a radius of 4. However, we need to check the endpoints, x = -4 and x = 4, to see if the series converges at these points.

Let's check the endpoints:

  • For x = 4: The series becomes ∑[n=0 to ∞] (7 * 4ⁿ) / (4ⁿ⁺¹) = ∑[n=0 to ∞] 7/4. This is a constant series that diverges because we're adding 7/4 infinitely many times.

  • For x = -4: The series becomes ∑[n=0 to ∞] (7 * (-4)ⁿ) / (4ⁿ⁺¹) = ∑[n=0 to ∞] 7 * (-1)ⁿ / 4. This series also diverges because it oscillates between positive and negative values and doesn't approach a finite limit.

Therefore, the interval of convergence for the power series is -4 < x < 4, or in interval notation, (-4, 4). We use parentheses because the series diverges at the endpoints.

Wrapping Up

So, we've successfully found the power series representation for f(x) = 7/(4-x), which is ∑[n=0 to ∞] (7xⁿ) / (4ⁿ⁺¹), and we've determined its interval of convergence to be (-4, 4). This whole process demonstrates the power of power series in representing functions and the importance of understanding convergence. By manipulating the function into a form suitable for the geometric series formula and applying the ratio test, we were able to achieve our goal.

Remember, this is just one example, and the world of power series is vast and full of fascinating applications. Keep exploring, keep questioning, and keep learning! You've got this!