Test questions on representing functions as power series using 1/(1-x)

📚 Quick Study Guide

• 🔢 Representing functions as power series often involves manipulating them into the form of a geometric series, which is based on $\frac{1}{1-x}$.
• 💡 The geometric series formula is: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ for $|x| < 1$.
• 🔄 To represent a function $f(x)$ as a power series, try to rewrite it in the form $a \cdot \frac{1}{1-g(x)}$, where $a$ is a constant and $g(x)$ is a function of $x$.
• 📝 Once you have the form $a \cdot \frac{1}{1-g(x)}$, substitute $g(x)$ for $x$ in the geometric series formula: $f(x) = a \cdot \sum_{n=0}^{\infty} [g(x)]^n$.
• 🎯 Determine the interval of convergence by finding the values of $x$ for which $|g(x)| < 1$.
• ➕ Differentiation and integration can be applied to power series within their interval of convergence to find power series representations of related functions.

🧪 Practice Quiz

1. Question 1: Which of the following is the power series representation of $\frac{1}{1+x}$?
   1. $\sum_{n=0}^{\infty} x^n$
   2. $\sum_{n=0}^{\infty} (-1)^n x^n$
   3. $\sum_{n=0}^{\infty} x^{2n}$
   4. $\sum_{n=0}^{\infty} (-1)^n x^{2n}$
2. Question 2: Find the power series representation of $\frac{1}{1-3x}$.
   1. $\sum_{n=0}^{\infty} x^n$
   2. $\sum_{n=0}^{\infty} 3^n x^n$
   3. $\sum_{n=0}^{\infty} (3x)^n$
   4. $\sum_{n=0}^{\infty} 3x^n$
3. Question 3: What is the interval of convergence for the series representation of $\frac{1}{1+x}$?
   1. $(-1, 1)$
   2. $[-1, 1]$
   3. $(- \infty, \infty)$
   4. $(-1, 1]$
4. Question 4: Find the power series for $\frac{x}{1-x}$.
   1. $\sum_{n=0}^{\infty} x^n$
   2. $\sum_{n=0}^{\infty} x^{n+1}$
   3. $\sum_{n=1}^{\infty} x^{n-1}$
   4. $\sum_{n=1}^{\infty} x^n$
5. Question 5: Determine the power series representation for $\frac{1}{2-x}$.
   1. $\sum_{n=0}^{\infty} \frac{x^n}{2^n}$
   2. $\sum_{n=0}^{\infty} \frac{x^n}{2^{n+1}}$
   3. $\sum_{n=0}^{\infty} \frac{x^{n+1}}{2^n}$
   4. $\sum_{n=0}^{\infty} \frac{x^{n+1}}{2^{n+1}}$
6. Question 6: What is the radius of convergence for the power series of $\frac{1}{2-x}$?
   1. 1
   2. 2
   3. 0.5
   4. $\infty$
7. Question 7: Represent $\frac{1}{1-x^2}$ as a power series.
   1. $\sum_{n=0}^{\infty} x^n$
   2. $\sum_{n=0}^{\infty} x^{2n}$
   3. $\sum_{n=0}^{\infty} 2x^n$
   4. $\sum_{n=0}^{\infty} 2x^{2n}$