Power Series Review Quiz: Definition, Radius, and Interval of Convergence

📚 Quick Study Guide

🔍 Power Series Definition: A power series centered at $a$ is an infinite series of the form $\sum_{n=0}^{\infty} c_n (x-a)^n$, where $c_n$ are coefficients and $a$ is the center.
• 📏 Radius of Convergence (R): Determines the interval for which the power series converges. Found using the Ratio Test or Root Test.
• 💡 Ratio Test: Given a series $\sum a_n$, compute $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. If $L < 1$, the series converges; if $L > 1$, the series diverges; if $L = 1$, the test is inconclusive.
• 🧮 Root Test: Given a series $\sum a_n$, compute $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. If $L < 1$, the series converges; if $L > 1$, the series diverges; if $L = 1$, the test is inconclusive.
• 🧭 Interval of Convergence: The set of all $x$ values for which the power series converges. It's given by $(a-R, a+R)$, $[a-R, a+R)$, $(a-R, a+R]$, or $[a-R, a+R]$, and endpoint behavior must be checked separately.
• 📝 Common Power Series: Some well-known power series include the geometric series $\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ for $|x| < 1$, and the Taylor series for $e^x$, $\sin x$, and $\cos x$.

Practice Quiz

1. What is the general form of a power series centered at $a$?
   1. $\sum_{n=0}^{\infty} c_n x^n$
   2. $\sum_{n=1}^{\infty} c_n (x-a)^n$
   3. $\sum_{n=0}^{\infty} c_n (x-a)^n$
   4. $\sum_{n=-\infty}^{\infty} c_n (x-a)^n$
2. Which test is commonly used to determine the radius of convergence of a power series?
   1. Integral Test
   2. Ratio Test
   3. Comparison Test
   4. Divergence Test
3. What does the radius of convergence, $R$, represent?
   1. The exact values for which the power series converges.
   2. Half the length of the interval of convergence.
   3. The point around which the series is centered.
   4. The limit of the series as $n$ approaches infinity.
4. If the radius of convergence is $R = 0$, what does this imply about the convergence of the power series?
   1. The series converges for all $x$.
   2. The series converges only at the center.
   3. The series diverges for all $x$.
   4. The series converges for $|x| < 1$.
5. What must be checked after finding the radius of convergence to determine the interval of convergence?
   1. The derivative of the power series.
   2. The endpoints of the interval.
   3. The center of the power series.
   4. The second derivative of the power series.
6. For the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$, what is the radius of convergence?
   1. $R = 0$
   2. $R = 1$
   3. $R = e$
   4. $R = \infty$
7. If a power series converges at $x = 3$ and diverges at $x = 7$, what can be said about its radius of convergence centered at $x = 0$?
   1. $R \leq 3$
   2. $R \geq 3$
   3. $R \leq 7$
   4. $R \geq 7$