The Formula for Finding the Taylor Series

Oct 13, 2025 by pinkandtrash.com 83 views

Introduction

In mathematics, the Taylor series is a powerful tool used to approximate functions as an infinite sum of terms. It is a fundamental concept in calculus and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the Taylor series of the function $$f(x)=\frac{1}{1-x}$$ centered at $$c=3$$. We will explore the concept of Taylor series, the formula for finding the Taylor series, and then apply it to the given function.

What is a Taylor Series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms. It is a way to approximate a function by expressing it as a sum of its derivatives evaluated at a specific point. The Taylor series is named after the English mathematician James Gregory and the Scottish mathematician James Gregory's friend, Brook Taylor.

The Formula for Finding the Taylor Series

The formula for finding the Taylor series of a function $$f(x)$$ centered at $$c$$ is given by:

f(x) = f(c) + \frac{f'(c)}{1!}(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \frac{f'''(c)}{3!}(x-c)^3 + \cdots

where $$f'(c)$$, $$f''(c)$$, and $$f'''(c)$$ are the first, second, and third derivatives of $$f(x)$$ evaluated at $$c$$, respectively.

Finding the Taylor Series of $$f(x)=\frac{1}{1-x}$$

To find the Taylor series of $$f(x)=\frac{1}{1-x}$$ centered at $$c=3$$, we need to find the derivatives of $$f(x)$$ evaluated at $$c=3$$.

First Derivative

The first derivative of $$f(x)$$ is given by:

f'(x) = \frac{1}{(1-x)^2}

Evaluating the first derivative at $$c=3$$, we get:

f'(3) = \frac{1}{(1-3)^2} = \frac{1}{4}

Second Derivative

The second derivative of $$f(x)$$ is given by:

f''(x) = \frac{2}{(1-x)^3}

Evaluating the second derivative at $$c=3$$, we get:

f''(3) = \frac{2}{(1-3)^3} = -\frac{2}{8} = -\frac{1}{4}

Third Derivative

The third derivative of $$f(x)$$ is given by:

f'''(x) = \frac{6}{(1-x)^4}

Evaluating the third derivative at $$c=3$$, we get:

f'''(3) = \frac{6}{(1-3)^4} = \frac{6}{16} = \frac{3}{8}

Higher-Order Derivatives

We can continue to find higher-order derivatives of $$f(x)$$ and evaluate them at $$c=3$$. However, we can observe a pattern in the derivatives:

f^{(n)}(x) = (-1)^{n+1} \frac{(n+1)!}{(1-x)^{n+1}}

Evaluating the higher-order derivatives at $$c=3$$, we get:

f^{(n)}(3) = (-1)^{n+1} \frac{(n+1)!}{(1-3)^{n+1}} = (-1)^{n+1} \frac{(n+1)!}{4^{n+1}}

Taylor Series

Now that we have found the derivatives of $$f(x)$$ evaluated at $$c=3$$, we can write the Taylor series of $$f(x)$$ centered at $$c=3$$:

f(x) = f(3) + \frac{f'(3)}{1!}(x-3) + \frac{f''(3)}{2!}(x-3)^2 + \frac{f'''(3)}{3!}(x-3)^3 + \cdots

Substituting the values of the derivatives, we get:

f(x) = \frac{1}{1-3} + \frac{1/4}{1!}(x-3) - \frac{1/4}{2!}(x-3)^2 + \frac{3/8}{3!}(x-3)^3 + \cdots

Simplifying the expression, we get:

f(x) = -\frac{1}{2} + \frac{1}{4}(x-3) - \frac{1}{8}(x-3)^2 + \frac{1}{16}(x-3)^3 + \cdots

Conclusion

In this article, we have found the Taylor series of the function $$f(x)=\frac{1}{1-x}$$ centered at $$c=3$$. We have used the formula for finding the Taylor series and evaluated the derivatives of the function at $$c=3$$. The Taylor series is given by:

f(x) = -\frac{1}{2} + \frac{1}{4}(x-3) - \frac{1}{8}(x-3)^2 + \frac{1}{16}(x-3)^3 + \cdots

This Taylor series can be used to approximate the function $$f(x)$$ for values of $$x$$ close to $$c=3$$.

Answer

The correct Taylor series of $$f(x)=\frac{1}{1-x}$$ centered at $$c=3$$ is:

\frac{1}{1-x}=\sum_{n=0}^{\infty}(-1)^{n+1} \frac{(n+1)!}{4^{n+1}}(x-3)^n

Introduction

In our previous article, we explored the concept of Taylor series and found the Taylor series of the function $$f(x)=\frac{1}{1-x}$$ centered at $$c=3$$. In this article, we will answer some frequently asked questions about Taylor series and provide additional insights into this powerful mathematical tool.

Q: What is the Taylor series of a function?

A: The Taylor series of a function $$f(x)$$ centered at $$c$$ is a mathematical representation of the function as an infinite sum of terms. It is a way to approximate the function by expressing it as a sum of its derivatives evaluated at the point $$c$$.

Q: How do I find the Taylor series of a function?

A: To find the Taylor series of a function $$f(x)$$ centered at $$c$$, you need to find the derivatives of the function evaluated at $$c$$ and then use the formula for the Taylor series:

f(x) = f(c) + \frac{f'(c)}{1!}(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \frac{f'''(c)}{3!}(x-c)^3 + \cdots

Q: What is the difference between a Taylor series and a Maclaurin series?

A: A Taylor series is a mathematical representation of a function as an infinite sum of terms, centered at a specific point $$c$$. A Maclaurin series is a special case of a Taylor series, where the center of the series is at $$c=0$$. In other words, a Maclaurin series is a Taylor series centered at $$c=0$$.

Q: Can I use the Taylor series to approximate a function?

A: Yes, you can use the Taylor series to approximate a function. The Taylor series is a powerful tool for approximating functions, especially for values of the input variable close to the center of the series.

Q: How accurate is the Taylor series approximation?

A: The accuracy of the Taylor series approximation depends on the number of terms included in the series. The more terms you include, the more accurate the approximation will be.

Q: Can I use the Taylor series to find the value of a function at a specific point?

A: Yes, you can use the Taylor series to find the value of a function at a specific point. By substituting the value of the input variable into the Taylor series, you can find the value of the function at that point.

Q: What are some common applications of Taylor series?

A: Taylor series have numerous applications in various fields, including:

Physics: Taylor series are used to describe the motion of objects, the behavior of electrical circuits, and the properties of materials.
Engineering: Taylor series are used to design and analyze systems, such as bridges, buildings, and electronic circuits.
Economics: Taylor series are used to model economic systems, including the behavior of markets and the effects of policy changes.
Computer Science: Taylor series are used in algorithms for solving differential equations and in the design of numerical methods for solving problems.

Conclusion

In this article, we have answered some frequently asked questions about Taylor series and provided additional insights into this powerful mathematical tool. We have also explored the concept of Taylor series and found the Taylor series of the function $$f(x)=\frac{1}{1-x}$$ centered at $$c=3$$. We hope that this article has been helpful in understanding the concept of Taylor series and its applications.

Frequently Asked Questions

Q: What is the Taylor series of a function?
A: The Taylor series of a function $$f(x)$$ centered at $$c$$ is a mathematical representation of the function as an infinite sum of terms.
Q: How do I find the Taylor series of a function?
A: To find the Taylor series of a function $$f(x)$$ centered at $$c$$, you need to find the derivatives of the function evaluated at $$c$$ and then use the formula for the Taylor series.
Q: What is the difference between a Taylor series and a Maclaurin series?
A: A Taylor series is a mathematical representation of a function as an infinite sum of terms, centered at a specific point $$c$$. A Maclaurin series is a special case of a Taylor series, where the center of the series is at $$c=0$$.
Q: Can I use the Taylor series to approximate a function?
A: Yes, you can use the Taylor series to approximate a function.
Q: How accurate is the Taylor series approximation?
A: The accuracy of the Taylor series approximation depends on the number of terms included in the series.
Q: Can I use the Taylor series to find the value of a function at a specific point?
A: Yes, you can use the Taylor series to find the value of a function at a specific point.

Glossary

Taylor series: A mathematical representation of a function as an infinite sum of terms, centered at a specific point $$c$$.
Maclaurin series: A special case of a Taylor series, where the center of the series is at $$c=0$$.
Derivative: A measure of how a function changes as its input variable changes.
Approximation: A way to estimate the value of a function at a specific point.
Accuracy: The degree to which an approximation is close to the actual value of the function.