High school calculus problems: finding the radius of convergence ratio test solutions

📚 Quick Study Guide

🔍 Ratio Test: The ratio test determines the convergence or divergence of an infinite series. For a series $\sum a_n$, calculate $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. 🧪 If $L < 1$, the series converges absolutely. 📈 If $L > 1$, the series diverges. 〰️ If $L = 1$, the test is inconclusive. 🧮 Power Series: A power series is a series of the form $\sum c_n(x-a)^n$, where $c_n$ are coefficients, $x$ is a variable, and $a$ is the center. 📍 Radius of Convergence (R): The radius of convergence is a non-negative real number or $\infty$ that represents the radius of the interval in which a power series converges. If $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = |x-a|/R$, then solve for $R$. Specifically, if $L = K|x-a|$ for some constant K, then $R = 1/K$. 💡 Finding R using the Ratio Test: Apply the ratio test to the power series. The radius of convergence $R$ is found by setting the limit $L < 1$ and solving for $|x-a| < R$.

Practice Quiz

1. Question 1: What is the general form of a power series?
1. A. $\sum c_n x^n$
2. B. $\sum c_n(x-a)^n$
3. C. $\sum c_n n!$
4. D. $\sum (x-a)^n$
2. Question 2: According to the Ratio Test, if $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = 2$, what can you conclude about the series?
1. A. The series converges absolutely.
2. B. The series diverges.
3. C. The test is inconclusive.
4. D. The series converges conditionally.
3. Question 3: Consider the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. What is the limit used in the Ratio Test to find the radius of convergence?
1. A. $\lim_{n \to \infty} |\frac{x}{n+1}|$
2. B. $\lim_{n \to \infty} |\frac{x}{n}|$
3. C. $\lim_{n \to \infty} |\frac{x^n}{n!}|$
4. D. $\lim_{n \to \infty} |\frac{1}{n!}|$
4. Question 4: If, after applying the ratio test to a power series, you find that $L = |x-3|$, what is the radius of convergence $R$?
1. A. $R = 1$
2. B. $R = 3$
3. C. $R = \infty$
4. D. $R = 0$
5. Question 5: What happens if the limit $L$ in the Ratio Test equals 1?
1. A. The series converges absolutely.
2. B. The series diverges.
3. C. The test is inconclusive.
4. D. The series converges conditionally.
6. Question 6: For the series $\sum_{n=1}^{\infty} n x^n$, what is the radius of convergence?
1. A. $R = 0$
2. B. $R = 1$
3. C. $R = \infty$
4. D. $R = 2$
7. Question 7: Consider the power series $\sum_{n=0}^{\infty} (2x)^n$. What is its radius of convergence?
1. A. $R = 1$
2. B. $R = 2$
3. C. $R = 1/2$
4. D. $R = \infty$