What is the Limit Comparison Test for Series?

📚 Quick Study Guide

• 🔍 The Limit Comparison Test (LCT) is used to determine the convergence or divergence of an infinite series by comparing it to another series whose convergence or divergence is known.
• 💡 Suppose we have two series, $\sum a_n$ and $\sum b_n$, where $a_n > 0$ and $b_n > 0$ for all $n$.
• 📝 Calculate the limit: $L = \lim_{n \to \infty} \frac{a_n}{b_n}$.
• 🧪 If $0 < L < \infty$, then both series either converge or both diverge.
• 📈 If $L = 0$ and $\sum b_n$ converges, then $\sum a_n$ also converges.
• 📉 If $L = \infty$ and $\sum b_n$ diverges, then $\sum a_n$ also diverges.

Practice Quiz

1. Question 1: What is the primary purpose of the Limit Comparison Test?
   1. (A) To find the exact sum of a series.
   2. (B) To determine if a series converges or diverges.
   3. (C) To find the derivative of a series.
   4. (D) To integrate a series.
2. Question 2: Under what condition can the Limit Comparison Test be applied to two series $\sum a_n$ and $\sum b_n$?
   1. (A) When $a_n$ and $b_n$ are both negative.
   2. (B) When $a_n$ and $b_n$ are alternating.
   3. (C) When $a_n > 0$ and $b_n > 0$ for all $n$.
   4. (D) When $a_n$ and $b_n$ are complex numbers.
3. Question 3: What is the conclusion if $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ and $0 < L < \infty$?
   1. (A) $\sum a_n$ converges and $\sum b_n$ diverges.
   2. (B) $\sum a_n$ diverges and $\sum b_n$ converges.
   3. (C) Both series either converge or both diverge.
   4. (D) Neither series converges nor diverges.
4. Question 4: 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) $\sum a_n$ neither converges nor diverges.
   4. (D) The test is inconclusive.
5. Question 5: If $\lim_{n \to \infty} \frac{a_n}{b_n} = \infty$ and $\sum b_n$ diverges, what can be concluded about $\sum a_n$?
   1. (A) $\sum a_n$ converges.
   2. (B) $\sum a_n$ diverges.
   3. (C) $\sum a_n$ neither converges nor diverges.
   4. (D) The test is inconclusive.
6. Question 6: Which series would be best to compare with $\sum \frac{1}{n^2 + 1}$ using the Limit Comparison Test?
   1. (A) $\sum \frac{1}{n}$
   2. (B) $\sum \frac{1}{n^2}$
   3. (C) $\sum 1$
   4. (D) $\sum \frac{1}{\sqrt{n}}$
7. Question 7: Consider $\sum_{n=1}^{\infty} \frac{3n^2 + 2n}{n^4 + n + 1}$. Which series is most suitable for comparison using the Limit Comparison Test?
   1. (A) $\sum \frac{1}{n}$
   2. (B) $\sum \frac{1}{n^2}$
   3. (C) $\sum \frac{3}{n}$
   4. (D) $\sum \frac{3}{n^2}$