Difference Between the Divergence Test and Other Series Convergence Tests

📚 Quick Study Guide

• 🔍 The Divergence Test: This test is your first line of defense. It's simple: if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.
• ⚠️ Limitation: The Divergence Test can only prove divergence. If $\lim_{n \to \infty} a_n = 0$, the test is inconclusive, and you need other tests.
• 🔢 Integral Test: Use this for series where $a_n = f(n)$ and $f(x)$ is a positive, continuous, and decreasing function for $x \geq 1$. Compare the series to the integral $\int_1^{\infty} f(x) dx$.
• ⚖️ Comparison Tests (Direct and Limit): These work by comparing your series to a known convergent or divergent series (e.g., geometric series, p-series).
• ➕ Alternating Series Test: Specifically for alternating series of the form $\sum (-1)^n b_n$ or $\sum (-1)^{n+1} b_n$. Check if $b_n > 0$, $b_n$ is decreasing, and $\lim_{n \to \infty} b_n = 0$.
• ➗ Ratio Test: Good for series involving factorials or exponentials. Calculate $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. If $L < 1$, the series converges; if $L > 1$, it diverges; if $L = 1$, the test is inconclusive.
• ⧿ Root Test: Similar to the Ratio Test, but calculate $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. Convergence, divergence, and inconclusive results follow the same rules as the Ratio Test.

Practice Quiz

1. Which of the following is the primary purpose of the Divergence Test?
   1. A. To determine if a series converges.
   2. B. To determine if a series diverges.
   3. C. To find the sum of a convergent series.
   4. D. To approximate the sum of a divergent series.
2. If $\lim_{n \to \infty} a_n = 0$, what can you conclude using the Divergence Test about the series $\sum a_n$?
   1. A. The series converges.
   2. B. The series diverges.
   3. C. The test is inconclusive.
   4. D. The series converges absolutely.
3. Which test is most suitable for determining the convergence or divergence of the series $\sum \frac{1}{n^2}$?
   1. A. Divergence Test
   2. B. Alternating Series Test
   3. C. Integral Test
   4. D. Ratio Test
4. For what type of series is the Alternating Series Test specifically designed?
   1. A. Series with only positive terms.
   2. B. Series with only negative terms.
   3. C. Series with terms that alternate in sign.
   4. D. Series with terms that are all zero.
5. When should you consider using the Ratio Test?
   1. A. When the terms of the series are decreasing slowly.
   2. B. When the series involves factorials or exponentials.
   3. C. When the terms alternate in sign.
   4. D. When the series is a p-series.
6. What is the conclusion if, using the Ratio Test, $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = 1$?
   1. A. The series converges.
   2. B. The series diverges.
   3. C. The test is inconclusive.
   4. D. The series converges absolutely.
7. Which test involves comparing a series to an integral?
   1. A. Divergence Test
   2. B. Ratio Test
   3. C. Integral Test
   4. D. Alternating Series Test