Find A Power Series Representation For The Function. Give Your Power Series Representation Centered At $x=0$.$ f(x) = \frac{6}{1-x^2} F(x) = \sum_{n=0}^{\infty}(\square) $

Introduction

In this article, we will explore the process of finding a power series representation for a given rational function. The function we will be working with is $f(x) = \frac{6}{1-x^2}$. Our goal is to express this function as a power series centered at $x=0$. This involves using the concept of geometric series and manipulating the function to obtain a power series representation.

Understanding Power Series

A power series is a mathematical representation of a function as an infinite sum of terms, each term being a power of the variable. The general form of a power series is:

$$f(x) = \sum_{n=0}^{\infty}a_n(x-c)^n$$

where $a_n$ are the coefficients of the power series, $c$ is the center of the power series, and $x$ is the variable.

Finding the Power Series Representation

To find the power series representation of the function $f(x) = \frac{6}{1-x^2}$, we can start by expressing the denominator as a geometric series. We know that the geometric series is given by:

$$\frac{1}{1-r} = \sum_{n=0}^{\infty}r^n$$

We can rewrite the denominator of the function as:

$$1-x^2 = (1-x)(1+x)$$

Now, we can use the geometric series formula to express the denominator as:

$$\frac{1}{1-x^2} = \frac{1}{(1-x)(1+x)} = \frac{1}{1-x} \cdot \frac{1}{1+x}$$

Using the geometric series formula, we can express each of these fractions as a power series:

$$\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$$

$$\frac{1}{1+x} = \sum_{n=0}^{\infty}(-x)^n$$

Now, we can multiply these two power series together to obtain the power series representation of the function:

$$f(x) = \frac{6}{1-x^2} = 6 \cdot \sum_{n=0}^{\infty}x^n \cdot \sum_{n=0}^{\infty}(-x)^n$$

Using the distributive property, we can expand this product:

$$f(x) = 6 \cdot \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} x^n (-x)^m$$

Now, we can simplify this expression by combining like terms:

$$f(x) = 6 \cdot \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} (-1)^m x^{n+m}$$

Simplifying the Power Series

To simplify the power series, we can use the fact that the sum of two infinite series is equal to the sum of the individual series. We can rewrite the power series as:

$$f(x) = 6 \cdot \sum_{k=0}^{\infty} \left( \sum_{n=0}^{k} (-1)^n \right) x^k$$

Now, we can evaluate the inner sum:

$$\sum_{n=0}^{k} (-1)^n = \begin{cases} 1 & \text{if } k \text{ is even} \\ -1 & \text{if } k \text{ is odd} \end{cases}$$

Using this result, we can simplify the power series:

$$f(x) = 6 \cdot \sum_{k=0}^{\infty} x^k \cdot \begin{cases} 1 & \text{if } k \text{ is even} \\ -1 & \text{if } k \text{ is odd} \end{cases}$$

Final Power Series Representation

After simplifying the power series, we can express the function $f(x) = \frac{6}{1-x^2}$ as:

$$f(x) = \sum_{n=0}^{\infty} 6 \cdot \begin{cases} 1 & \text{if } n \text{ is even} \\ -1 & \text{if } n \text{ is odd} \end{cases} x^n$$

This is the power series representation of the function centered at $x=0$.

Conclusion

In this article, we have found the power series representation of the function $f(x) = \frac{6}{1-x^2}$ centered at $x=0$. We have used the concept of geometric series and manipulated the function to obtain a power series representation. The final power series representation is given by:

$$f(x) = \sum_{n=0}^{\infty} 6 \cdot \begin{cases} 1 & \text{if } n \text{ is even} \\ -1 & \text{if } n \text{ is odd} \end{cases} x^n$$

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Introduction

In our previous article, we explored the process of finding a power series representation for a given rational function. We used the concept of geometric series and manipulated the function to obtain a power series representation. In this article, we will answer some common questions related to power series representation of rational functions.

Q: What is the power series representation of a rational function?

A: The power series representation of a rational function is an infinite sum of terms, each term being a power of the variable. The general form of a power series is:

$$f(x) = \sum_{n=0}^{\infty}a_n(x-c)^n$$

where $a_n$ are the coefficients of the power series, $c$ is the center of the power series, and $x$ is the variable.

Q: How do I find the power series representation of a rational function?

A: To find the power series representation of a rational function, you can use the concept of geometric series and manipulate the function to obtain a power series representation. You can start by expressing the denominator as a geometric series and then use the distributive property to expand the product.

Q: What is the center of the power series?

A: The center of the power series is the value of $x$ around which the power series is centered. In our previous article, we found the power series representation of the function $f(x) = \frac{6}{1-x^2}$ centered at $x=0$.

Q: How do I determine the coefficients of the power series?

A: To determine the coefficients of the power series, you can use the concept of geometric series and manipulate the function to obtain a power series representation. You can also use the fact that the sum of two infinite series is equal to the sum of the individual series.

Q: What is the significance of the power series representation of a rational function?

A: The power series representation of a rational function is significant because it allows us to approximate the function for values of $x$ close to the center of the power series. This is useful in many applications, such as physics, engineering, and computer science.

Q: Can I use the power series representation of a rational function to find the derivative of the function?

A: Yes, you can use the power series representation of a rational function to find the derivative of the function. You can differentiate each term in the power series to obtain the derivative of the function.

Q: Can I use the power series representation of a rational function to find the integral of the function?

A: Yes, you can use the power series representation of a rational function to find the integral of the function. You can integrate each term in the power series to obtain the integral of the function.

Q: What are some common applications of power series representation of rational functions?

A: Some common applications of power series representation of rational functions include:

• Physics: Power series representation of rational functions is used to describe the behavior of physical systems, such as the motion of a pendulum or the vibration of a spring.
• Engineering: Power series representation of rational functions is used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
• Computer Science: Power series representation of rational functions is used in computer graphics, game development, and other areas of computer science.

Conclusion

In this article, we have answered some common questions related to power series representation of rational functions. We have discussed the power series representation of a rational function, how to find the power series representation, the center of the power series, the coefficients of the power series, and some common applications of power series representation of rational functions.