Solved examples: Representing functions as power series from 1/(1-x)

📚 Quick Study Guide

• 🔢 Geometric Series: The foundation is the geometric series formula: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$, valid for $|x| < 1$.
• 🔄 Substitution: Replace $x$ with a function of $x$ (e.g., $x^2$, $-3x$) to obtain new power series representations. Remember to adjust the interval of convergence.
• ➗ Algebraic Manipulation: Sometimes, you need to manipulate the given function algebraically to get it into the form $\frac{A}{1-f(x)}$, where $A$ is a constant and $f(x)$ is a function of $x$.
• ➕ Differentiation and Integration: Differentiating or integrating a power series term-by-term results in a new power series with related convergence properties.
• 📏 Interval of Convergence: Always determine the interval of convergence for the resulting power series. Use ratio test or other convergence tests.
• 💡 Example: To represent $\frac{1}{1+x^2}$ as a power series, substitute $-x^2$ for $x$ in the geometric series formula: $\frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-x^2)^n = \sum_{n=0}^{\infty} (-1)^n x^{2n}$.

🧪 Practice Quiz

1. Question 1: What is the power series representation of $\frac{1}{1-4x}$?
   1. $\sum_{n=0}^{\infty} x^{4n}$
   2. $\sum_{n=0}^{\infty} 4^n x^n$
   3. $\sum_{n=0}^{\infty} 4x^n$
   4. $\sum_{n=0}^{\infty} (4x)^n$
2. Question 2: Find the power series for $\frac{x}{1-x^3}$.
   1. $\sum_{n=0}^{\infty} x^{3n}$
   2. $\sum_{n=0}^{\infty} x^{3n+1}$
   3. $\sum_{n=0}^{\infty} x^{n+3}$
   4. $\sum_{n=0}^{\infty} 3x^n$
3. Question 3: What is the interval of convergence for the series $\sum_{n=0}^{\infty} (2x)^n$?
   1. $(-1, 1)$
   2. $[-\frac{1}{2}, \frac{1}{2}]$
   3. $(-\frac{1}{2}, \frac{1}{2})$
   4. $(-2, 2)$
4. Question 4: Express $\frac{1}{1+9x^2}$ as a power series.
   1. $\sum_{n=0}^{\infty} (9x^2)^n$
   2. $\sum_{n=0}^{\infty} (-1)^n (9x^2)^n$
   3. $\sum_{n=0}^{\infty} (-9x^2)^n$
   4. $\sum_{n=0}^{\infty} 9^n x^{2n}$
5. Question 5: Determine the power series representation of $\frac{2}{1-x}$.
   1. $\sum_{n=0}^{\infty} 2x^n$
   2. $2\sum_{n=0}^{\infty} x^n$
   3. $\sum_{n=0}^{\infty} (2x)^n$
   4. $\sum_{n=0}^{\infty} x^{2n}$
6. Question 6: Find the power series of $\frac{1}{1-(x-2)}$.
   1. $\sum_{n=0}^{\infty} (x-2)^n$
   2. $\sum_{n=0}^{\infty} x^n - 2$
   3. $\sum_{n=0}^{\infty} x^n - \sum_{n=0}^{\infty} 2^n$
   4. $\sum_{n=0}^{\infty} x^n - 2^n$
7. Question 7: What is the power series representation of $\frac{1}{5-x}$?
   1. $\sum_{n=0}^{\infty} \frac{x^n}{5^n}$
   2. $\sum_{n=0}^{\infty} \frac{x^n}{5}$
   3. $\sum_{n=0}^{\infty} \frac{x^n}{5^{n+1}}$
   4. $\frac{1}{5}\sum_{n=0}^{\infty} \frac{x^n}{5^n}$