Unit 10.3 – The nth-Term Test for Divergence BC ONLY

AP® Calculus BC | Test for Divergence

Why This Matters: The nth-Term Test for Divergence (also called the Divergence Test) is the FIRST test you should always check! It's the quickest and easiest test, but has a critical limitation: it can ONLY prove divergence, never convergence. Understanding what this test can and cannot do is essential for all series work!

🎯 The nth-Term Test for Divergence

The Divergence Test

THE TEST STATEMENT:

For the series \(\sum_{n=1}^{\infty} a_n\):

\[ \text{If } \lim_{n \to \infty} a_n \neq 0, \text{ then } \sum_{n=1}^{\infty} a_n \text{ DIVERGES} \]

⚠️ CRITICAL LIMITATION: If \(\lim_{n \to \infty} a_n = 0\), the test is INCONCLUSIVE. The series might converge OR diverge. You need a different test!

🔄 The Contrapositive Form

Necessary Condition for Convergence

CONTRAPOSITIVE (Equivalent Statement):

\[ \text{If } \sum_{n=1}^{\infty} a_n \text{ converges, then } \lim_{n \to \infty} a_n = 0 \]

What This Means:

Terms going to zero is NECESSARY for convergence
But NOT SUFFICIENT! Terms can go to zero and series still diverges
Example: Harmonic series \(\sum \frac{1}{n}\) has terms → 0 but diverges

📊 Decision Logic

How to Use the nth-Term Test:

Step 1: Calculate \(\lim_{n \to \infty} a_n\)

Step 2: Check the result

If limit ≠ 0: Series DIVERGES ✓ (Test complete!)
If limit = 0: INCONCLUSIVE (Need another test)
If limit DNE (oscillates): Series DIVERGES ✓

📝 Remember: This test is like a security guard—it can kick out (diverge) series that clearly don't belong, but can't approve entry (convergence) for anyone!

✅❌ What the Test Can and Cannot Prove

Test Capabilities
Situation	Result	Conclusion
\(\lim_{n \to \infty} a_n \neq 0\)	DIVERGES	✓ Definitive answer
\(\lim_{n \to \infty} a_n = \infty\)	DIVERGES	✓ Definitive answer
Limit DNE (oscillates)	DIVERGES	✓ Definitive answer
\(\lim_{n \to \infty} a_n = 0\)	INCONCLUSIVE	⚠️ Need another test

📖 Comprehensive Worked Examples

Example 1: Test Shows Divergence

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

Solution:

Apply nth-term test:

\[ \lim_{n \to \infty} \frac{n}{n+1} = \lim_{n \to \infty} \frac{1}{1 + 1/n} = 1 \neq 0 \]

CONCLUSION: Series DIVERGES by nth-term test

Example 2: Test is Inconclusive (Series Actually Converges)

Problem: What about \(\sum_{n=1}^{\infty} \frac{1}{n^2}\)?

Check the limit:

\[ \lim_{n \to \infty} \frac{1}{n^2} = 0 \]

nth-term test is INCONCLUSIVE. Need another test!

(This series actually converges—we'll learn p-series test later)

Example 3: Test is Inconclusive (Series Actually Diverges)

Problem: Analyze \(\sum_{n=1}^{\infty} \frac{1}{n}\) (Harmonic Series)

Check the limit:

\[ \lim_{n \to \infty} \frac{1}{n} = 0 \]

nth-term test is INCONCLUSIVE

BUT: This is the famous harmonic series—it DIVERGES!

Shows that \(\lim a_n = 0\) doesn't guarantee convergence!

Example 4: Limit is Infinity

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

Calculate limit:

\[ \lim_{n \to \infty} n^2 = \infty \neq 0 \]

DIVERGES by nth-term test (limit ≠ 0)

Example 5: Oscillating Limit

Problem: Analyze \(\sum_{n=1}^{\infty} (-1)^n\)

Check limit:

\[ \lim_{n \to \infty} (-1)^n \text{ does not exist (oscillates between -1 and 1)} \]

DIVERGES by nth-term test (limit DNE, so ≠ 0)

Example 6: More Complex Limit

Problem: Does \(\sum_{n=1}^{\infty} \frac{n^2 + 3n}{2n^2 + 1}\) converge?

Evaluate limit:

\[ \lim_{n \to \infty} \frac{n^2 + 3n}{2n^2 + 1} = \lim_{n \to \infty} \frac{1 + 3/n}{2 + 1/n^2} = \frac{1}{2} \neq 0 \]

DIVERGES by nth-term test

⭐ Famous Examples to Remember

Classic Series:

1. Harmonic Series (terms → 0 but DIVERGES)

\[ \sum_{n=1}^{\infty} \frac{1}{n}: \quad \lim_{n \to \infty} \frac{1}{n} = 0 \text{ (inconclusive)} \]

Series DIVERGES (need integral test to prove)

2. Geometric Series with |r| < 1 (terms → 0 and CONVERGES)

\[ \sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n: \quad \lim_{n \to \infty} \left(\frac{1}{2}\right)^n = 0 \text{ (inconclusive)} \]

Series CONVERGES (geometric test proves it)

3. Divergent Series (terms don't → 0)

\[ \sum_{n=1}^{\infty} \frac{n}{n+1}: \quad \lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0 \]

DIVERGES by nth-term test ✓

💡 Essential Tips & Strategies

✅ Success Strategies:

ALWAYS check this test first: It's the quickest!
Remember: can only prove divergence: Not convergence
If limit = 0: Say "inconclusive" and move to another test
If limit ≠ 0: You're done—series diverges
Use L'Hôpital's if needed: For indeterminate forms
Don't say "converges" when limit = 0: Big mistake!
Simplify before taking limit: Makes calculation easier
Watch for oscillating limits: DNE means ≠ 0

🔥 Quick Checklist:

Find \(\lim_{n \to \infty} a_n\)
If limit ≠ 0 or DNE → DIVERGES (done!)
If limit = 0 → INCONCLUSIVE (try another test)
NEVER conclude convergence from this test alone

❌ Common Mistakes to Avoid

Mistake 1: Saying series converges when \(\lim a_n = 0\) (WRONG! Inconclusive!)
Mistake 2: Not recognizing test is inconclusive when limit = 0
Mistake 3: Forgetting to check this test first (easiest test to apply)
Mistake 4: Confusing sequence convergence with series convergence
Mistake 5: Wrong limit calculation (algebra errors)
Mistake 6: Not simplifying before taking limit
Mistake 7: Thinking limit = 0 is sufficient for convergence
Mistake 8: Not stating "by nth-term test" in conclusion
Mistake 9: Saying test "fails" instead of "inconclusive"
Mistake 10: Not recognizing when limit DNE (oscillates) means divergence

📝 Practice Problems

Use the nth-term test (if possible):

\(\sum_{n=1}^{\infty} \frac{2n}{3n+1}\)
\(\sum_{n=1}^{\infty} \frac{1}{2^n}\)
\(\sum_{n=1}^{\infty} \cos(n\pi)\)
\(\sum_{n=1}^{\infty} \frac{\ln n}{n}\)
\(\sum_{n=1}^{\infty} \frac{n!}{2^n}\)

Answers:

DIVERGES (limit = 2/3 ≠ 0)
INCONCLUSIVE (limit = 0; actually converges by geometric)
DIVERGES (limit DNE, oscillates)
INCONCLUSIVE (limit = 0; actually diverges)
DIVERGES (limit = ∞ ≠ 0)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

State you're using nth-term test: "By the nth-term test..."
Show limit calculation: \(\lim_{n \to \infty} a_n = ...\)
State conclusion clearly: "Since limit ≠ 0, series diverges"
If limit = 0: Say "inconclusive" or "test fails" or "need another test"
NEVER say: "Converges because limit = 0"
Show work: Don't just state the limit
Use proper notation: \(\lim_{n \to \infty}\)
Be explicit: "≠ 0" or "= 0"

💯 Exam Strategy:

Always try this test first (takes 5 seconds!)
Calculate \(\lim_{n \to \infty} a_n\)
If limit ≠ 0: Write "Diverges by nth-term test" and move on
If limit = 0: Write "nth-term test inconclusive, trying [other test]"
Show all limit work clearly
Never claim convergence from this test

⚡ Quick Reference Guide

NTH-TERM TEST ESSENTIALS

The Test:

\[ \text{If } \lim_{n \to \infty} a_n \neq 0 \text{, then } \sum a_n \text{ DIVERGES} \]

Decision Tree:

\(\lim a_n \neq 0\) → DIVERGES ✓
\(\lim a_n = 0\) → INCONCLUSIVE ⚠️
Limit DNE → DIVERGES ✓

Contrapositive:

If \(\sum a_n\) converges → \(\lim a_n = 0\)

(Terms → 0 is NECESSARY but NOT SUFFICIENT)

Remember:

Can ONLY prove divergence!
Always check this test first!
If limit = 0, test is inconclusive!

Master the nth-Term Test! The nth-term test for divergence states: if \(\lim_{n\to\infty} a_n \neq 0\) (or DNE), then \(\sum a_n\) DIVERGES. Critical limitation: if \(\lim a_n = 0\), test is INCONCLUSIVE—series might converge or diverge, need another test. Contrapositive: convergent series must have terms → 0 (necessary condition), but this is NOT sufficient—harmonic series \(\sum \frac{1}{n}\) has terms → 0 yet diverges! Strategy: always check this test FIRST (quickest); if limit ≠ 0, done (diverges); if limit = 0, move to another test. Common error: claiming convergence when limit = 0 (WRONG!). This test is like a bouncer—can reject (diverge) but can't approve (converge). Use on every series problem as first step. Master the distinction between "diverges" vs "inconclusive"—crucial for BC exams! 🎯✨