Write the Taylor series for 𝑓(𝑥) = 1/𝑥 centered at 𝑎 = 1

To write the Taylor series for \( f(x) = \frac{1}{x} \) centered at \( a = 1 \), we first need to find the derivatives of \( f(x) \) and evaluate them at \( x = 1 \).

The Taylor series for a function \( f(x) \) centered at \( a \) is given by:
\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \]

For \( f(x) = \frac{1}{x} \):

1. \( f(x) = \frac{1}{x} \)
2. \( f'(x) = -\frac{1}{x^2} \)
3. \( f''(x) = \frac{2}{x^3} \)
4. \( f'''(x) = -\frac{6}{x^4} \)
5. \( f^{(4)}(x) = \frac{24}{x^5} \)
6. And so on...

Evaluating these derivatives at \( x = 1 \):

1. \( f(1) = 1 \)
2. \( f'(1) = -1 \)
3. \( f''(1) = 2 \)
4. \( f'''(1) = -6 \)
5. \( f^{(4)}(1) = 24 \)

Now, we can write the Taylor series centered at \( a = 1 \):

\[ f(x) = 1 - (x-1) + \frac{2}{2!}(x-1)^2 - \frac{6}{3!}(x-1)^3 + \frac{24}{4!}(x-1)^4 - \cdots \]

Simplifying the coefficients:

\[ f(x) = 1 - (x-1) + (x-1)^2 - (x-1)^3 + (x-1)^4 - \cdots \]

Thus, the Taylor series for \( f(x) = \frac{1}{x} \) centered at \( a = 1 \) is:

\[ f(x) = \sum_{n=0}^{\infty} (-1)^n (x-1)^n \]

This series converges for \( |x-1| < 1 \).

Do you want more details or have any questions?

Here are 5 more questions you might find interesting:

1. What is the radius of convergence for the Taylor series of \( e^x \) centered at \( x = 0 \)?
2. How do you find the Maclaurin series for \( \sin(x) \)?
3. Can you derive the Taylor series for \( \ln(x) \) centered at \( x = 1 \)?
4. What is the Taylor series for \( \cos(x) \) centered at \( x = \pi/2 \)?
5. How do you determine the interval of convergence for a given power series?

**Tip:** When calculating derivatives for Taylor series, it can be helpful to look for patterns in the derivatives to simplify the process.

Learn how to write the Taylor series for the function f(x) = 1/x centered at a = 1. Includes detailed steps on finding derivatives, evaluating them, and constructing the Taylor series. Suitable for advanced college students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation