Direct Comparison Test vs. Limit Comparison Test explained

📚 Quick Study Guide

• 🔍 Direct Comparison Test: Used to determine the convergence or divergence of a series by comparing it to another series whose convergence or divergence is known.
• ➕ If $0 \le a_n \le b_n$ for all $n$, and $\sum b_n$ converges, then $\sum a_n$ converges.
• ➖ If $0 \le b_n \le a_n$ for all $n$, and $\sum b_n$ diverges, then $\sum a_n$ diverges.
• 📈 Limit Comparison Test: Compares the limit of the ratio of two series' terms to determine convergence or divergence.
• ➗ If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$, where $0 < c < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.
• 💡 Choose a series $\sum b_n$ that is similar to $\sum a_n$ but simpler to analyze (e.g., p-series, geometric series).
• 📝 Both tests require careful selection of the comparison series $\sum b_n$.

Practice Quiz

1. Which of the following is a necessary condition for using the Direct Comparison Test?
   1. A. The terms of the series must be alternating.
   2. B. The terms of both series must be positive.
   3. C. The series must be a geometric series.
   4. D. The limit of the terms must be zero.
2. If $0 \le a_n \le b_n$ for all $n$, and $\sum_{n=1}^{\infty} b_n$ converges, what can you conclude about $\sum_{n=1}^{\infty} a_n$?
   1. A. $\sum a_n$ diverges.
   2. B. $\sum a_n$ converges.
   3. C. $\sum a_n$ is conditionally convergent.
   4. D. No conclusion can be made.
3. Which test is most suitable for series where the ratio of terms is easier to compute than the terms themselves?
   1. A. Direct Comparison Test
   2. B. Integral Test
   3. C. Limit Comparison Test
   4. D. Alternating Series Test
4. For the Limit Comparison Test, if $\lim_{n \to \infty} \frac{a_n}{b_n} = 0$, and $\sum b_n$ converges, what can be concluded about $\sum a_n$?
   1. A. $\sum a_n$ diverges.
   2. B. $\sum a_n$ converges.
   3. C. $\sum a_n$ may converge or diverge, test inconclusive.
   4. D. $\sum a_n$ oscillates.
5. When is the Direct Comparison Test inconclusive?
   1. A. When $a_n < b_n$ and $\sum b_n$ diverges.
   2. B. When $a_n > b_n$ and $\sum b_n$ converges.
   3. C. When the limit of $a_n/b_n$ is 1.
   4. D. The Direct Comparison Test is never inconclusive.
6. Which of the following series would be best suited for Limit Comparison Test with the series $\sum_{n=1}^{\infty} \frac{1}{n^2 + 3n + 2}$?
   1. A. $\sum_{n=1}^{\infty} \frac{1}{n}$
   2. B. $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   3. C. $\sum_{n=1}^{\infty} \frac{1}{2^n}$
   4. D. $\sum_{n=1}^{\infty} 1$
7. What is the key difference in the application of the Direct Comparison Test and the Limit Comparison Test?
   1. A. Direct Comparison Test requires finding a specific inequality, while Limit Comparison Test requires finding a limit.
   2. B. Limit Comparison Test can only be used for geometric series.
   3. C. Direct Comparison Test is always more accurate than Limit Comparison Test.
   4. D. Limit Comparison Test only applies to alternating series.