Understanding the Ratio Test for AP Calculus

📚 Quick Study Guide

🔍 The Ratio Test helps determine the convergence or divergence of an infinite series. ➕ Consider the series $\sum a_n$. ➗ Compute the limit: $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. ✅ If $L < 1$, the series converges absolutely. ❌ If $L > 1$ (or $L = \infty$), the series diverges. ❓ If $L = 1$, the test is inconclusive; you need another method. 💡 Remember to use absolute values to handle alternating series.

Practice Quiz

1. What is the first step in applying the Ratio Test to the series $\sum_{n=1}^{\infty} \frac{n^2}{2^n}$?
   1. Find the derivative of the terms.
   2. Compute the limit of the absolute value of the ratio of consecutive terms.
   3. Integrate the terms.
   4. Check if the terms are decreasing.


2. If, after applying the Ratio Test, you find that $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = 2$, what can you conclude about the series $\sum a_n$?
   1. The series converges absolutely.
   2. The series converges conditionally.
   3. The series diverges.
   4. The test is inconclusive.


3. For what type of series is the Ratio Test most useful?
   1. Geometric series
   2. Alternating series
   3. Series with factorials or exponential terms
   4. p-series


4. Evaluate the limit: $L = \lim_{n \to \infty} |\frac{(n+1)!}{n!}|$
   1. 0
   2. 1
   3. $\infty$
   4. -1


5. Apply the Ratio Test to the series $\sum_{n=1}^{\infty} \frac{1}{n^3}$. What does the test conclude?
   1. The series converges.
   2. The series diverges.
   3. The test is inconclusive.
   4. The series oscillates.


6. Consider the series $\sum_{n=1}^{\infty} (-1)^n \frac{n}{n^2+1}$. What should you do before applying the Ratio Test?
   1. Ignore the $(-1)^n$ term.
   2. Take the absolute value of the terms.
   3. Find the derivative of the terms.
   4. Integrate the terms.


7. If $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = 1$, what is the next step to determine the convergence or divergence of the series?
   1. Conclude the series converges.
   2. Conclude the series diverges.
   3. Apply a different convergence test.
   4. Stop, as the series neither converges nor diverges.