Illustrative examples of radius and interval of convergence after power series operations.

📚 Quick Study Guide

• ➕ Addition/Subtraction: If two power series, $\sum a_n(x-c)^n$ and $\sum b_n(x-c)^n$, have radii of convergence $R_1$ and $R_2$ respectively, then the radius of convergence of their sum or difference is at least $\min(R_1, R_2)$. The interval of convergence must be checked at the endpoints.
• ✖️ Multiplication: The radius of convergence of the product of two power series is at least $\min(R_1, R_2)$, similar to addition and subtraction. Finding the exact interval can be complex.
• ➗ Division: For the quotient of two power series, the radius of convergence is determined by the zeros of the denominator. If $\frac{f(x)}{g(x)}$ is the quotient, look for where $g(x) = 0$ to find the radius.
• ⚙️ Differentiation/Integration: Differentiation and integration of a power series do not change the radius of convergence, but the interval of convergence may change because the endpoints must be checked.
• 📝 Important Note: When performing operations, always check the endpoints of the resulting interval to determine the exact interval of convergence.

Practice Quiz

1. Question 1: Suppose $\sum a_n x^n$ has a radius of convergence $R = 3$ and $\sum b_n x^n$ has a radius of convergence $R = 5$. What is the minimum radius of convergence of $\sum (a_n + b_n) x^n$?
   1. 2
   2. 3
   3. 5
   4. 8
2. Question 2: If the power series $\sum a_n (x-2)^n$ converges for $-1 < x < 5$, what is its radius of convergence?
   1. 2
   2. 3
   3. 4
   4. 5
3. Question 3: Given $\sum_{n=0}^{\infty} \frac{x^n}{2^n}$, what is the interval of convergence?
   1. (-2, 2)
   2. [-2, 2]
   3. (-2, 2]
   4. [-2, 2)
4. Question 4: If a power series $\sum a_n x^n$ has radius of convergence $R$, what is the radius of convergence of its derivative, $\sum n a_n x^{n-1}$?
   1. $R-1$
   2. $R+1$
   3. $R$
   4. Cannot be determined
5. Question 5: If $\sum a_n x^n$ converges for $|x| < 4$, for what values of $x$ does $\sum a_n (x-1)^n$ converge?
   1. $-4 < x < 4$
   2. $-3 < x < 5$
   3. $-5 < x < 3$
   4. $-4 < x-1 < 4$
6. Question 6: Suppose $f(x) = \sum_{n=0}^{\infty} x^n$. What is the radius of convergence of $f(x^2)$?
   1. 0
   2. 0.5
   3. 1
   4. 2
7. Question 7: If $\sum a_n x^n$ has radius of convergence $R = 1$, what is the radius of convergence of $\sum a_n (2x)^n$?
   1. 0.5
   2. 1
   3. 2
   4. 4