https://file.aiquickdraw.com/mathbot/file/1729929859778zv28hp5o.jpg

To find the power series representation of \( \frac{1}{1 + 8x} \) centered at \( x = 3 \), we can follow these steps:

1. **Rewrite the function:** Start by rewriting \( \frac{1}{1 + 8x} \) in a form that resembles the geometric series formula. We want it in the form \( \frac{1}{1 - u} \), where \( |u| < 1 \).

   \[
   \frac{1}{1 + 8x} = \frac{1}{1 - (-8x)}
   \]

   This setup means we can use the geometric series formula:
   
   \[
   \frac{1}{1 - u} = \sum_{n=0}^{\infty} u^n
   \]

2. **Express in terms of \( x - 3 \):** Since we want the series centered at \( x = 3 \), we let \( u = -8(x - 3) - 24 \)

Which of the following is the power series representation centred at x = 3 for the function 1/(1 + 8x)?

Learn how to find the power series representation of 1/(1 + 8x) centered at x = 3. This example involves algebraic manipulation and the geometric series expansion. Suitable for undergraduate students studying calculus or mathematical analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation