Study Guide: The Divergence and Integral Tests

Learning Objectives

After reviewing this study guide, you should be able to:

• Use the divergence test to determine whether an infinite series converges or diverges.
• Verify the conditions and use the integral test to determine the convergence of a series.
• Identify $p-series and evaluate their convergence based on the value of p.
• Estimate the value of a series by finding bounds on its remainder term.

Key Terms & Glossary

• Infinite Series: The sum of the terms of an infinite sequence.
• Partial Sum: The sum of the first n terms of a series, denoted as S_n. A series converges if the sequence of its partial sums converges.
• Divergence Test (nth-Term Test): A preliminary test stating that if the sequence of terms does not approach 0, the series diverges.
• Integral Test: A method comparing a discrete infinite sum to a continuous improper integral to prove convergence or divergence.
• Harmonic Series: The specific divergent series \sum_{n=1}^{\infty} \frac{1}{n}$.
• **$p-series**: A family of series in the form \sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p$ is a real number.

The "Big Idea"

A central problem in integral calculus is determining if an infinitely long list of positive numbers adds up to a finite total (convergence) or grows to infinity (divergence). Calculating the exact limit of infinite partial sums is often impossible.

Instead, we use logical tests. The Divergence Test is your first line of defense: if the individual terms don't eventually shrink to zero, the sum will naturally blow up to infinity. If they do shrink to zero, the test is inconclusive, and we move to heavier tools like the Integral Test. By mapping discrete sequence terms to continuous functions, the Integral Test allows us to use our knowledge of improper integrals (like calculating the area under a curve to infinity) to definitively prove whether the discrete sum converges or diverges.

Formula / Concept Box

[!WARNING] The #1 Trap: The Divergence Test can only prove divergence. If \lim_{n \to \infty} a_n = 0, you know absolutely nothing. The test is inconclusive. Do NOT conclude the series converges!

Hierarchical Outline

• 1. The Divergence Test
  • Connecting series convergence to the sequence of Partial Sums (S_n).
  • Core theorem: If a series converges, \lim a_n = 0.
  • The contrapositive (Divergence Test): If \lim a_n \neq 0, the series diverges.
• 2. The Integral Test
  • Pre-requisite conditions for the function f(x)$:
    • Positive ($f(x) > 0)
    • Continuous (no breaks or vertical asymptotes)
    • Decreasing (f'(x) < 0)
  • The mechanism: Comparing rectangle area sums (a_n) to the area under the curve (\int f(x)dx$).
• 3. The $p-Series
  • Definition of a p-series.
  • Using the Integral Test to prove the rule: converges for p>1$, diverges for$p\le1$.
  • The Harmonic Series as the ultimate boundary case for divergence.

Visual Anchors

Series Testing Decision Flow

Geometric Intuition of the Integral Test

Definition-Example Pairs

• Divergence Test: A quick check verifying if the terms of a series eventually become zero.
  • Example: For $\sum \frac{n}{n+1}$, $\lim_{n \to \infty} \frac{n}{n+1} = 1$. Since $1 \neq 0, the series diverges immediately.
• Integral Test: A test mapping a discrete sequence a_n to a continuous function f(x) to check the convergence of the improper integral.
  • Example: For \sum \frac{1}{n^2}$, evaluate$\int_1^\infty \frac{1}{x^2} dx$. The integral yields 1 (it converges), meaning the series converges too.
• **$p-Series**: A series whose terms are the reciprocal of n raised to a constant power p$.
  • Example: $\sum \frac{1}{n^3}$ is a $p$-series with $p=3$. Because $3 > 1$, the series converges.
• Harmonic Series: The most famous divergent $p$-series, where $p=1$.
  • Example: $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$ Even though the terms eventually reach zero, they don't shrink fast enough to prevent the sum from reaching infinity.

Worked Examples

Example 1: Applying the Divergence Test

Determine if the series $\sum_{n=1}^\infty \frac{5n^2}{2n^2 + 1}$ converges or diverges.

1. Identify the test: The terms are a rational function where the degree of the numerator matches the denominator. Start with the Divergence Test.
2. Calculate limit: $\lim_{n \to \infty} \frac{5n^2}{2n^2 + 1}$
3. Evaluate: Divide top and bottom by $n^2$ to get $\lim_{n \to \infty} \frac{5}{2 + \frac{1}{n^2}} = \frac{5}{2}$.
4. Conclusion: Since the limit is $\frac{5}{2}$, which is $\neq 0, the series diverges by the Divergence Test.

Example 2: Applying the Integral Test

Determine if the series \sum_{n=2}^\infty \frac{1}{n \ln(n)} converges or diverges.

1. Check conditions: Let f(x) = \frac{1}{x \ln(x)}$. For$x \ge 2$,$f(x)$ is positive, continuous, and decreasing (because the denominator strictly increases).
2. Set up the improper integral: $\int_2^\infty \frac{1}{x \ln(x)} dx$
3. Integrate: Use $u$-substitution with $u = \ln(x)$ and $du = \frac{1}{x} dx$. $\int \frac{1}{u} du = \ln|u| = \ln|\ln(x)|$
4. Evaluate limits: $\lim_{t \to \infty} \left[ \ln(\ln(t)) - \ln(\ln(2)) \right]$
5. Conclusion: As $t \to \infty$, $\ln(\ln(t))$ grows to $\infty. The integral diverges. Therefore, the series diverges by the Integral Test.

Checkpoint Questions

1. If you apply the Divergence Test and find that \lim_{n \to \infty} a_n = 0, what exact conclusion can you draw about the series \sum a_n?
2. What three specific conditions must a function f(x) satisfy before you can use the Integral Test?
3. Does the series \sum_{n=1}^\infty \frac{1}{\sqrt{n}} converge or diverge, and which specific test easily proves this?
4. Why is the Harmonic Series conceptually important when discussing the limitations of the Divergence Test?

▶Click to expand answers

1. None. The test is completely inconclusive. The series might converge or it might diverge; you must use a different test to find out.
2. The function f(x) must be positive, continuous, and decreasing on the interval [1, \infty).
3. It diverges. This can be rewritten as \sum \frac{1}{n^{1/2}}$. It is a$p$-series where$p = 0.5$. Because$p \le 1, the series diverges.
4. The Harmonic Series (\sum \frac{1}{n}) has terms that shrink to zero (\lim \frac{1}{n} = 0$), yet the series itself diverges. It is the definitive proof that terms approaching zero is a necessary, but not sufficient, condition for a series to converge.