Applying the Limit Comparison Test: Tips for Success

📚 Quick Study Guide

• 🔍 What is the Limit Comparison Test? 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.
• ➕ When to Use It: LCT is particularly useful when dealing with series that resemble p-series or geometric series.
• 📝 The Test: Suppose we have two series, $\sum a_n$ and $\sum b_n$, where $a_n > 0$ and $b_n > 0$ for all $n$.
  • If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$, where $0 < c < \infty$, then both series either converge or both diverge.
  • If $\lim_{n \to \infty} \frac{a_n}{b_n} = 0$ and $\sum b_n$ converges, then $\sum a_n$ converges.
  • If $\lim_{n \to \infty} \frac{a_n}{b_n} = \infty$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.
• 💡 Choosing a Comparison Series: Select a series $\sum b_n$ that behaves similarly to $\sum a_n$ for large $n$. Often, this involves focusing on the dominant terms of $a_n$.
• 🧮 Example: To analyze $\sum \frac{1}{n^2 + 1}$, compare it to $\sum \frac{1}{n^2}$ (a convergent p-series).

Practice Quiz

1. Which of the following is the correct condition for the Limit Comparison Test to conclude that both series $\sum a_n$ and $\sum b_n$ converge or diverge?
   1. $\lim_{n \to \infty} \frac{a_n}{b_n} = 0$
   2. $\lim_{n \to \infty} \frac{a_n}{b_n} = \infty$
   3. $\lim_{n \to \infty} \frac{a_n}{b_n} = c$, where $0 < c < \infty$
   4. $\lim_{n \to \infty} \frac{a_n}{b_n} = 1$
2. Consider the series $\sum_{n=1}^{\infty} \frac{1}{n^2 + n}$. What series would be most suitable to compare it to using the Limit Comparison Test?
   1. $\sum_{n=1}^{\infty} \frac{1}{n}$
   2. $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   3. $\sum_{n=1}^{\infty} 1$
   4. $\sum_{n=1}^{\infty} \frac{1}{2^n}$
3. If $\lim_{n \to \infty} \frac{a_n}{b_n} = 0$ and $\sum b_n$ diverges, what can you conclude about $\sum a_n$?
   1. $\sum a_n$ converges
   2. $\sum a_n$ diverges
   3. The test is inconclusive
   4. $\sum a_n$ oscillates
4. Determine the convergence of $\sum_{n=1}^{\infty} \frac{2n+1}{n^2+3n+1}$ using the Limit Comparison Test. Which comparison series is most appropriate?
   1. $\sum_{n=1}^{\infty} \frac{1}{n}$
   2. $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   3. $\sum_{n=1}^{\infty} 1$
   4. $\sum_{n=1}^{\infty} \frac{1}{2^n}$
5. If $\lim_{n \to \infty} \frac{a_n}{b_n} = \infty$ and $\sum b_n$ converges, what can you conclude about $\sum a_n$?
   1. $\sum a_n$ converges
   2. $\sum a_n$ diverges
   3. The test is inconclusive
   4. $\sum a_n$ oscillates
6. For the series $\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^2 + 1}$, which comparison series is best suited for the Limit Comparison Test?
   1. $\sum_{n=1}^{\infty} \frac{1}{n}$
   2. $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$
   3. $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   4. $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$
7. Suppose you are given $\sum a_n = \sum \frac{5n^2 - 3n}{n^4 + 1}$. Which series $\sum b_n$ would be most appropriate for comparison using the Limit Comparison Test?
   1. $\sum \frac{1}{n}$
   2. $\sum \frac{1}{n^2}$
   3. $\sum \frac{5}{n}$
   4. $\sum \frac{5}{n^2}$