Unit 10.4 – Integral Test for Convergence BC ONLY

AP® Calculus BC | Using Integrals to Test Series

Why This Matters: The Integral Test is one of the most powerful convergence tests because it connects series to improper integrals! It's especially useful for series with terms like \(\frac{1}{n^p}\), \(\frac{\ln n}{n}\), and \(\frac{1}{n\ln n}\). This test also gives us the p-series test, one of the most important results in series!

🎯 The Integral Test

The Integral Test Statement

THE TEST:

Let \(f(x)\) be a function such that:

\(f(x)\) is continuous on \([k, \infty)\)
\(f(x)\) is positive on \([k, \infty)\)
\(f(x)\) is decreasing on \([k, \infty)\)

And let \(a_n = f(n)\). Then:

\[ \sum_{n=k}^{\infty} a_n \text{ and } \int_k^{\infty} f(x) \, dx \]

BOTH converge or BOTH diverge

In Simple Terms:

If \(\int_k^{\infty} f(x) \, dx\) converges → Series converges
If \(\int_k^{\infty} f(x) \, dx\) diverges → Series diverges

📝 Key Insight: The integral and series don't have the same VALUE, but they have the same CONVERGENCE BEHAVIOR!

✅ Conditions for the Test

Three Required Conditions

1. Continuous:

\(f(x)\) must be continuous on \([k, \infty)\)

Usually satisfied for common functions like \(\frac{1}{x^p}\), \(\frac{1}{x \ln x}\), etc.

2. Positive:

\(f(x) > 0\) for all \(x \geq k\)

All terms must be positive

3. Decreasing:

\(f(x)\) is decreasing on \([k, \infty)\), meaning \(f'(x) < 0\)

How to check: Find \(f'(x)\) and verify it's negative for \(x \geq k\)

⚠️ IMPORTANT: You should VERIFY these conditions (especially decreasing) before applying the test on the AP exam!

⭐ The p-Series Test (Consequence of Integral Test)

The p-Series

DEFINITION:

\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \]

where \(p\) is a constant

Convergence Rule:

If \(p > 1\): Series CONVERGES

If \(p \leq 1\): Series DIVERGES

Special Cases to MEMORIZE:

\(p = 1\): \(\sum \frac{1}{n}\) = Harmonic series (DIVERGES)
\(p = 2\): \(\sum \frac{1}{n^2}\) CONVERGES
\(p = 1/2\): \(\sum \frac{1}{\sqrt{n}}\) DIVERGES
\(p = 3\): \(\sum \frac{1}{n^3}\) CONVERGES

📋 How to Apply the Integral Test

Step-by-Step Process:

Define \(f(x)\): Let \(f(x) = a_n\) with \(n\) replaced by \(x\)
Verify conditions: Check continuous, positive, and decreasing
Set up integral: \(\int_k^{\infty} f(x) \, dx\)
Evaluate as limit: \(\lim_{t \to \infty} \int_k^t f(x) \, dx\)
Determine convergence:
- Finite limit → Series converges
- Infinite limit → Series diverges

📖 Comprehensive Worked Examples

Example 1: Using Integral Test (Converges)

Problem: Does \(\sum_{n=1}^{\infty} \frac{1}{n^3}\) converge?

Solution:

Step 1: Identify this as p-series with \(p = 3\)

Since \(p = 3 > 1\), series CONVERGES by p-series test!

Alternative: Use Integral Test directly

Let \(f(x) = \frac{1}{x^3}\)

Continuous, positive, decreasing for \(x \geq 1\) ✓

\[ \int_1^{\infty} \frac{1}{x^3} \, dx = \lim_{t \to \infty} \int_1^t x^{-3} \, dx \]

\[ = \lim_{t \to \infty} \left[-\frac{1}{2x^2}\right]_1^t = \lim_{t \to \infty} \left(-\frac{1}{2t^2} + \frac{1}{2}\right) = \frac{1}{2} \]

Integral converges → Series CONVERGES

Example 2: Harmonic Series (Diverges)

Problem: Prove \(\sum_{n=1}^{\infty} \frac{1}{n}\) diverges using integral test.

Setup:

Let \(f(x) = \frac{1}{x}\)

Continuous, positive, decreasing for \(x \geq 1\) ✓

Evaluate integral:

\[ \int_1^{\infty} \frac{1}{x} \, dx = \lim_{t \to \infty} \int_1^t \frac{1}{x} \, dx = \lim_{t \to \infty} [\ln x]_1^t \]

\[ = \lim_{t \to \infty} (\ln t - \ln 1) = \lim_{t \to \infty} \ln t = \infty \]

Integral diverges → Series DIVERGES

Example 3: More Complex Function

Problem: Does \(\sum_{n=2}^{\infty} \frac{1}{n \ln n}\) converge?

Set up:

Let \(f(x) = \frac{1}{x \ln x}\)

For \(x \geq 2\): continuous ✓, positive ✓, decreasing ✓

Evaluate integral:

\[ \int_2^{\infty} \frac{1}{x \ln x} \, dx \]

Use substitution: \(u = \ln x\), \(du = \frac{1}{x}dx\)

\[ = \lim_{t \to \infty} \int_{\ln 2}^{\ln t} \frac{1}{u} \, du = \lim_{t \to \infty} [\ln u]_{\ln 2}^{\ln t} \]

\[ = \lim_{t \to \infty} (\ln(\ln t) - \ln(\ln 2)) = \infty \]

Integral diverges → Series DIVERGES

Example 4: Verifying Decreasing

Problem: Test \(\sum_{n=1}^{\infty} \frac{n}{n^2 + 1}\)

Check if decreasing:

Let \(f(x) = \frac{x}{x^2 + 1}\)

\[ f'(x) = \frac{(x^2+1)(1) - x(2x)}{(x^2+1)^2} = \frac{1 - x^2}{(x^2+1)^2} \]

For \(x \geq 1\): \(1 - x^2 < 0\), so \(f'(x) < 0\) ✓ Decreasing!

Apply integral test:

\[ \int_1^{\infty} \frac{x}{x^2+1} \, dx = \frac{1}{2}\int_1^{\infty} \frac{2x}{x^2+1} \, dx \]

Let \(u = x^2 + 1\):

\[ = \frac{1}{2} \lim_{t \to \infty} [\ln(x^2+1)]_1^t = \frac{1}{2} \lim_{t \to \infty} \ln(t^2+1) = \infty \]

Series DIVERGES

📊 Quick p-Series Reference

p-Series Convergence
Series	p value	Converges?
\(\sum \frac{1}{n}\)	\(p = 1\)	NO (Harmonic)
\(\sum \frac{1}{\sqrt{n}}\)	\(p = 1/2\)	NO
\(\sum \frac{1}{n^{3/2}}\)	\(p = 3/2\)	YES
\(\sum \frac{1}{n^2}\)	\(p = 2\)	YES
\(\sum \frac{1}{n^3}\)	\(p = 3\)	YES

💡 Essential Tips & Strategies

✅ Success Strategies:

Check for p-series first: If it's \(\sum \frac{1}{n^p}\), use p-series test!
Verify decreasing: Find \(f'(x) < 0\) or show algebraically
Start integral at same index: If series starts at n=2, use \(\int_2^{\infty}\)
Use substitution freely: Common for \(\ln\) terms
Remember: convergence only: Doesn't give the sum!
Starting index doesn't affect convergence: Can start at n=1, 2, etc.
Conditions matter: State continuous, positive, decreasing
Common integrals: Know \(\int \frac{1}{x} dx\) and \(\int x^{-p} dx\)

🔥 Recognition Patterns:

\(\sum \frac{1}{n^p}\): Use p-series test directly
\(\sum \frac{1}{n \ln n}\): Integral test (diverges)
\(\sum \frac{1}{n (\ln n)^p}\): Integral test (converges if \(p > 1\))
\(\sum \frac{1}{n^2 + 1}\): Compare to p-series or use integral test

❌ Common Mistakes to Avoid

Mistake 1: Not verifying function is decreasing before using test
Mistake 2: Confusing \(p > 1\) with \(p < 1\) for p-series
Mistake 3: Thinking integral value = series sum (only convergence matches!)
Mistake 4: Forgetting to take limit for improper integral
Mistake 5: Not checking all three conditions (continuous, positive, decreasing)
Mistake 6: Wrong bounds on integral (should match series starting point)
Mistake 7: Integration errors (especially with ln)
Mistake 8: Saying harmonic series converges (it diverges!)
Mistake 9: Not simplifying before integrating
Mistake 10: Forgetting absolute value doesn't matter (terms already positive)

📝 Practice Problems

Determine convergence using integral or p-series test:

\(\sum_{n=1}^{\infty} \frac{1}{n^{5/2}}\)
\(\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n}}\)
\(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}\)
\(\sum_{n=1}^{\infty} ne^{-n^2}\)
\(\sum_{n=1}^{\infty} \frac{\ln n}{n^2}\)

Answers:

CONVERGES (p = 5/2 > 1)
DIVERGES (p = 1/3 < 1)
CONVERGES (integral test with u-sub)
CONVERGES (integral test, use u = -n²)
CONVERGES (integral test or comparison)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

State the test: "By the integral test..." or "By p-series test..."
Verify conditions: "f(x) is continuous, positive, and decreasing for x ≥ 1"
Show integral setup: \(\int_k^{\infty} f(x) \, dx\)
Show limit: \(\lim_{t \to \infty} \int_k^t f(x) \, dx\)
Show integration work: Don't skip steps
Evaluate limit: State whether finite or infinite
State conclusion: "Integral converges, so series converges"
For p-series: Identify p and state rule

💯 Exam Strategy:

Check if it's a p-series first (quickest!)
If not p-series, consider integral test
Define f(x) by replacing n with x
Verify three conditions (at least mention them)
Set up improper integral
Evaluate as limit
Conclude based on convergence of integral

⚡ Quick Reference Guide

INTEGRAL TEST ESSENTIALS

The Test:

If f is continuous, positive, decreasing:

\[ \sum a_n \text{ and } \int f(x)\,dx \text{ have same convergence} \]

p-Series Test:

\[ \sum \frac{1}{n^p} \begin{cases} \text{CONVERGES} & \text{if } p > 1 \\ \text{DIVERGES} & \text{if } p \leq 1 \end{cases} \]

Conditions Required:

Continuous on \( This comprehensive HTML section includes: ✅ **Complete Coverage**: Integral test for convergence (BC only) ✅ **BC Badge**: Clear indication this is BC-only content ✅ **Test Statement**: Complete with all three conditions ✅ **Conditions Explained**: Continuous, positive, decreasing ✅ **p-Series Test**: As a major consequence with convergence rule ✅ **Step-by-Step Process**: How to apply the test ✅ **4 Worked Examples**: p-series, harmonic, complex functions, verifying decreasing ✅ **Comparison Table**: p-series quick reference ✅ **Tips & Tricks**: Success strategies and recognition patterns ✅ **Common Mistakes**: 10 detailed errors to avoid ✅ **Practice Problems**: 5 problems with complete answers ✅ **AP® Exam Tips**: What BC graders look for ✅ **Quick Reference**: All essential formulas ✅ **Proper MathJax**: Configuration and LaTeX rendering ✅ **Responsive Design**: Mobile-friendly ✅ **Clean Styling**: Purple BC theme, white background, bold fonts Ready to paste into WordPress! This provides complete coverage of the integral test, one of the most important convergence tests for BC students.