A Guide to Testing Endpoints for Power Series Interval of Convergence

📚 Quick Study Guide

🔍 Power Series: A power series centered at $c$ has the form $\sum_{n=0}^{\infty} a_n(x-c)^n$. 💡 Interval of Convergence: The interval of $x$ values for which the power series converges. 📝 Radius of Convergence (R): Determined by the Ratio Test or Root Test: $R = \lim_{n \to \infty} |\frac{a_n}{a_{n+1}}|$ or $R = \lim_{n \to \infty} \frac{1}{\sqrt[n]{|a_n|}}$ (if the limit exists). 🧮 Endpoints: Values $x = c \pm R$ must be tested separately for convergence. ➕ Testing Endpoints: Substitute $x = c \pm R$ into the power series. The resulting series will be a constant series which can be tested for convergence using tests like the Alternating Series Test, Comparison Test, or p-test. 📈 Alternating Series Test: If the alternating series $\sum_{n=1}^{\infty} (-1)^n b_n$ satisfies (1) $b_n > 0$, (2) $b_n$ is decreasing, and (3) $\lim_{n \to \infty} b_n = 0$, then the series converges. ⚖️ p-test: The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.

🧪 Practice Quiz

1. Which of the following is the general form of a power series centered at $c$?
1. $\sum_{n=0}^{\infty} a_n x^n$
2. $\sum_{n=0}^{\infty} a_n(x+c)^n$
3. $\sum_{n=0}^{\infty} a_n(x-c)^n$
4. $\sum_{n=1}^{\infty} a_n(x-c)^n$
2. What is the first step in finding the interval of convergence for a power series?
1. Test the endpoints.
2. Determine the radius of convergence.
3. Check for absolute convergence.
4. Apply the Alternating Series Test.
3. Given a power series with radius of convergence $R$ centered at $c$, what values should be tested as endpoints?
1. $c + R$ only
2. $c - R$ only
3. $c \pm R$
4. $R \pm c$
4. Which test is most suitable for checking convergence at endpoints that result in alternating series?
1. Ratio Test
2. Root Test
3. Alternating Series Test
4. p-test
5. If substituting an endpoint into a power series results in the series $\sum_{n=1}^{\infty} \frac{1}{n}$, does the series converge at that endpoint?
1. Yes, it converges absolutely.
2. Yes, it converges conditionally.
3. No, it diverges.
4. It cannot be determined.
6. Consider the power series $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n^2}$. What test should be used to determine convergence at the endpoints?
1. Ratio Test
2. Alternating Series Test
3. p-test
4. Root Test
7. If the power series $\sum_{n=0}^{\infty} a_n (x-c)^n$ converges at $x = c + R$, but diverges at $x = c - R$, what is the interval of convergence?
1. $(c - R, c + R)$
2. $[c - R, c + R)$
3. $(c - R, c + R]$
4. $[c - R, c + R]$