Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 18
Expand the quantity about 0 in terms of the variable given. Give four nonzero terms. \(\frac{1}{2+x}\) in terms of \(\frac{x}{2}\)
Short Answer
Expert verified
The expansion is \( \frac{1}{2} - \frac{x}{4} + \frac{x^2}{8} - \frac{x^3}{16} \).
Step by step solution
01
Set Up the Binomial Series Expansion
Identify the expression to expand. Write the expression as \( f(x) = \frac{1}{2+x} \). We want to expand this around \( x = 0 \) and express it in terms of \( \frac{x}{2} \). First, reformulate \( f(x) \) as follows: \( f(x) = \frac{1}{2} \cdot \frac{1}{1 + \frac{x}{2}} \).
02
Identify the Expansion Formula
Recognize that \( \frac{1}{1-z} = 1 + z + z^2 + z^3 + \cdots \) is the form for expanding \( \frac{1}{1-z} \) with \(|z| < 1\). Here, \(z = -\frac{x}{2}\).
03
Expand Using the Formula
Use the series expansion \( \frac{1}{1-z} = 1 + z + z^2 + z^3 + \cdots \) by substituting \( z = -\frac{x}{2} \). This gives \( \frac{1}{1 + \frac{x}{2}} \approx 1 - \frac{x}{2} + \left(\frac{x}{2}\right)^2 - \left(\frac{x}{2}\right)^3 + \cdots \).
04
Multiply to Complete the Expansion
Multiply the expanded series by \( \frac{1}{2} \):\[ \frac{1}{2} \left(1 - \frac{x}{2} + \frac{x^2}{4} - \frac{x^3}{8} + \cdots\right) = \frac{1}{2} - \frac{x}{4} + \frac{x^2}{8} - \frac{x^3}{16} + \cdots \]
05
Extract Four Nonzero Terms
Identify the first four non-zero terms from the expansion to be:\[ \frac{1}{2}, -\frac{x}{4}, \frac{x^2}{8}, -\frac{x^3}{16} \]
06
Write the Final Expanded Form
Summarize the expansion:\[ \frac{1}{2+x} \approx \frac{1}{2} - \frac{x}{4} + \frac{x^2}{8} - \frac{x^3}{16} \] in terms of \( \frac{x}{2} \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Taylor Series
The Taylor series is a powerful tool in mathematics that approximates functions using polynomials. Essentially, it expresses a function as an infinite sum of terms, calculated from the values of its derivatives at a single point. When we talk about Taylor series expansion, we refer to this kind of approximation.
In the context of the given exercise, \( f(x) = \frac{1}{2+x} \) is expanded using a Taylor-like approach in terms of \( \frac{x}{2} \), making it easier to manage.
- To use the Taylor series, we select a point 'a' where the function is going to be expanded around.
- The general form for the Taylor series is: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \]
- For our problem here, we are expanding around \( x = 0 \).
In the context of the given exercise, \( f(x) = \frac{1}{2+x} \) is expanded using a Taylor-like approach in terms of \( \frac{x}{2} \), making it easier to manage.
Polynomial Approximation
Polynomial approximation, as the term suggests, involves approximating a function using polynomials. This is highly useful because polynomials are simpler to work with analytically, computationally, and graphically.
- The goal is to find a polynomial that is "close" to your target function within a specific interval or at particular points.
- The coefficients of the polynomial are determined to minimize the difference between the target function and the polynomial across your interval.
- In this exercise, the target \( \frac{1}{2+x} \) is approximated by a polynomial of degree 3.
Power Series Expansion
A power series is a series of the form \( \sum\limits_{n=0}^{\infty} a_n (x-c)^n \), where \( a_n \) represents the coefficient of the nth term and \( c \) is the center of the series.
Power series expansion is particularly helpful for expanding functions around a certain point, simplifying them into terms that are easier to handle. In essence:
Power series expansion is particularly helpful for expanding functions around a certain point, simplifying them into terms that are easier to handle. In essence:
- A power series can represent various functions depending on the choice of coefficients \( a_n \).
- This series can often be truncated to include only a limited number of terms, providing an approximation to the function.
- In our context, the power series given by \( \frac{1}{2} \left( 1 - \frac{x}{2} + \left(\frac{x}{2}\right)^2 - \left(\frac{x}{2}\right)^3 + \cdots \right) \) provides a simplified version of \( f(x) \) close to \( x=0 \).
