Unit 10.8 – Ratio Test for Convergence BC ONLY

AP® Calculus BC | The Power Tool for Series with Factorials and Exponentials

Why This Matters: The Ratio Test is THE go-to test for series with factorials (\(n!\)), exponentials (\(a^n\)), or combinations of both! It's incredibly powerful and works when most other tests fail. The test compares consecutive terms—if the ratio gets smaller, series converges; if larger, it diverges. Master this and you can handle almost any series!

🎯 The Ratio Test

The Ratio Test Statement

THE TEST:

For the series \(\sum_{n=1}^{\infty} a_n\) with positive terms, let:

\[ L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n} \]

If \(L < 1\):

Series CONVERGES ABSOLUTELY

If \(L > 1\) or \(L = \infty\):

Series DIVERGES

If \(L = 1\):

Test is INCONCLUSIVE (try another test)

📝 Key Insight: The test measures how fast terms are shrinking. If each term is less than some fixed fraction of the previous term (L < 1), the series behaves like a geometric series and converges!

🤔 When to Use the Ratio Test

Best Scenarios for Ratio Test

USE Ratio Test when you see:

Factorials: \(n!\), \((2n)!\), etc.
Exponentials: \(a^n\), \(e^n\), \(2^n\), etc.
Combinations: \(\frac{n!}{2^n}\), \(\frac{3^n}{n!}\), etc.
Powers of n: \(n^n\), \(n^{2n}\), etc.
Products of these: Complex expressions

DON'T use Ratio Test for:

p-series: Use p-series test instead
Simple rational functions: Use comparison tests
When L = 1: Test fails (common with p-series!)

📋 How to Apply the Ratio Test

Step-by-Step Process:

Write \(a_n\): Identify the general term
Write \(a_{n+1}\): Replace every n with (n+1)
Form ratio: \(\frac{a_{n+1}}{a_n}\)
Simplify: Cancel common factors (crucial!)
Take limit: \(\lim_{n \to \infty} \frac{a_{n+1}}{a_n}\)
Compare to 1: L < 1, L > 1, or L = 1?
Conclude: Converges, diverges, or inconclusive

🔧 Factorial Simplification Tricks

Essential Factorial Rules

Basic Factorial Property:

\[ (n+1)! = (n+1) \cdot n! \]

Use this to cancel factorials!

Simplification Pattern:

\[ \frac{(n+1)!}{n!} = n+1 \]

\[ \frac{n!}{(n+1)!} = \frac{1}{n+1} \]

For Double Factorials:

\[ \frac{(2n+2)!}{(2n)!} = (2n+2)(2n+1) \]

💡 Pro Tip: Always expand the larger factorial just enough to cancel with the smaller one!

📖 Comprehensive Worked Examples

Example 1: Series with Factorial

Problem: Does \(\sum_{n=1}^{\infty} \frac{n!}{5^n}\) converge?

Solution using Ratio Test:

Step 1: Identify \(a_n\)

\[ a_n = \frac{n!}{5^n} \]

Step 2: Find \(a_{n+1}\)

\[ a_{n+1} = \frac{(n+1)!}{5^{n+1}} \]

Step 3: Form ratio and simplify

\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)!/5^{n+1}}{n!/5^n} = \frac{(n+1)!}{n!} \cdot \frac{5^n}{5^{n+1}} \]

\[ = (n+1) \cdot \frac{1}{5} = \frac{n+1}{5} \]

Step 4: Take limit

\[ L = \lim_{n \to \infty} \frac{n+1}{5} = \infty \]

Since \(L = \infty > 1\), series DIVERGES by Ratio Test

Example 2: Series with Exponential

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

Set up:

\[ a_n = \frac{2^n}{n!}, \quad a_{n+1} = \frac{2^{n+1}}{(n+1)!} \]

Compute ratio:

\[ \frac{a_{n+1}}{a_n} = \frac{2^{n+1}/(n+1)!}{2^n/n!} = \frac{2^{n+1}}{2^n} \cdot \frac{n!}{(n+1)!} = 2 \cdot \frac{1}{n+1} = \frac{2}{n+1} \]

Take limit:

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

Since \(L = 0 < 1\), series CONVERGES by Ratio Test

Example 3: Test is Inconclusive

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

Apply Ratio Test:

\[ a_n = \frac{1}{n^2}, \quad a_{n+1} = \frac{1}{(n+1)^2} \]

\[ \frac{a_{n+1}}{a_n} = \frac{1/(n+1)^2}{1/n^2} = \frac{n^2}{(n+1)^2} \]

\[ L = \lim_{n \to \infty} \frac{n^2}{(n+1)^2} = \lim_{n \to \infty} \frac{1}{(1+1/n)^2} = 1 \]

\(L = 1\) → Ratio Test is INCONCLUSIVE

(We know this is p-series with p=2, which converges, but Ratio Test can't tell us!)

Example 4: Complex Series

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

Setup:

\[ a_n = \frac{n! \cdot 3^n}{(2n)!}, \quad a_{n+1} = \frac{(n+1)! \cdot 3^{n+1}}{(2n+2)!} \]

Form ratio:

\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)! \cdot 3^{n+1}}{(2n+2)!} \cdot \frac{(2n)!}{n! \cdot 3^n} \]

\[ = \frac{(n+1)!}{n!} \cdot \frac{3^{n+1}}{3^n} \cdot \frac{(2n)!}{(2n+2)!} \]

\[ = (n+1) \cdot 3 \cdot \frac{1}{(2n+2)(2n+1)} = \frac{3(n+1)}{(2n+2)(2n+1)} \]

Simplify and take limit:

\[ = \frac{3(n+1)}{2(n+1)(2n+1)} = \frac{3}{2(2n+1)} \]

\[ L = \lim_{n \to \infty} \frac{3}{2(2n+1)} = 0 \]

\(L = 0 < 1\) → Series CONVERGES

Example 5: With Alternating Signs

Problem: Does \(\sum_{n=1}^{\infty} \frac{(-1)^n \cdot 5^n}{n!}\) converge?

Apply Ratio Test to absolute values:

\[ a_n = \left|\frac{(-1)^n \cdot 5^n}{n!}\right| = \frac{5^n}{n!} \]

\[ \frac{a_{n+1}}{a_n} = \frac{5^{n+1}/(n+1)!}{5^n/n!} = \frac{5}{n+1} \]

\[ L = \lim_{n \to \infty} \frac{5}{n+1} = 0 < 1 \]

Series converges ABSOLUTELY by Ratio Test

📊 Ratio Test Decision Guide

ptionion>Interpreting the Limit L

Limit Value	Conclusion	Notes
\(L < 1\)	CONVERGES	Absolute convergence
\(L > 1\)	DIVERGES	Terms don't go to zero
\(L = \infty\)	DIVERGES	Terms growing
\(L = 1\)	INCONCLUSIVE	Try another test

💡 Essential Tips & Strategies

✅ Success Strategies:

Perfect for factorials: Ratio Test is THE tool for n!
Simplify before limit: Cancel everything possible
Expand factorials minimally: Just enough to cancel
For exponentials: \(\frac{a^{n+1}}{a^n} = a\)
Absolute values for alternating: Test \(|a_n|\)
L = 1 is boundary: Test fails here
Show all steps: Don't skip simplification on exams
State test name: "By Ratio Test..."

🔥 Quick Patterns:

\(\frac{n!}{a^n}\): Usually diverges if a is constant
\(\frac{a^n}{n!}\): Always converges (exponential < factorial)
\(\frac{n^k}{a^n}\): Converges if a > 1 (exponential wins)
\(\frac{a^n}{n^k}\): Diverges if a > 1 (exponential wins)

❌ Common Mistakes to Avoid

Mistake 1: Not simplifying ratio before taking limit
Mistake 2: Expanding factorial incorrectly
Mistake 3: Forgetting \(L = 1\) is INCONCLUSIVE (not divergent!)
Mistake 4: Using Ratio Test on p-series (always gives L=1)
Mistake 5: Not using absolute values for alternating series
Mistake 6: Algebra errors when simplifying complex ratios
Mistake 7: Writing \(\frac{a_{n+1}}{a_n}\) backwards
Mistake 8: Not stating conclusion clearly
Mistake 9: Trying to find the sum (test only gives convergence!)
Mistake 10: Missing absolute value bars when needed

📝 Practice Problems

Use the Ratio Test:

\(\sum_{n=1}^{\infty} \frac{3^n}{n!}\)
\(\sum_{n=1}^{\infty} \frac{n!}{10^n}\)
\(\sum_{n=1}^{\infty} \frac{n^2 \cdot 2^n}{3^n}\)
\(\sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!}\)
\(\sum_{n=1}^{\infty} \frac{(-1)^n n^n}{n!}\)

Answers:

CONVERGES (L = 0)
DIVERGES (L = ∞)
CONVERGES (L = 2/3 < 1)
CONVERGES (L = 0)
DIVERGES (L = ∞, by Stirling or direct calculation)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

State "Ratio Test": Name the test explicitly
Write \(a_n\) and \(a_{n+1}\): Show both terms
Form ratio clearly: \(\frac{a_{n+1}}{a_n}\)
Show simplification: All cancellation steps
Show limit calculation: \(\lim_{n \to \infty}\)
State limit value: L = [value]
Compare to 1: "Since L < 1..."
State conclusion: "Series converges by Ratio Test"

💯 Exam Strategy:

Look for factorials or exponentials (use Ratio Test!)
Write out \(a_n\) clearly
Write out \(a_{n+1}\) (replace all n with n+1)
Form fraction \(\frac{a_{n+1}}{a_n}\)
Simplify completely (cancel factorials, combine exponentials)
Take limit as n → ∞
Compare to 1 and conclude
If L = 1, state inconclusive and try another test

⚡ Quick Reference Guide

RATIO TEST ESSENTIALS

The Test:

\[ L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n} \]

\(L < 1\) → CONVERGES
\(L > 1\) → DIVERGES
\(L = 1\) → INCONCLUSIVE

Best For:

Factorials: n!, (2n)!
Exponentials: \(a^n\), \(e^n\)
Combinations: \(\frac{n!}{2^n}\)

Key Simplifications:

\(\frac{(n+1)!}{n!} = n+1\)
\(\frac{a^{n+1}}{a^n} = a\)
Cancel common factors!

Remember:

Simplify BEFORE taking limit!
L = 1 means test FAILS!
Perfect for factorials!

Master the Ratio Test! The Ratio Test compares consecutive terms: \(L = \lim_{n\to\infty} \frac{a_{n+1}}{a_n}\). If \(L < 1\) → converges absolutely; if \(L > 1\) or \(L = \infty\) → diverges; if \(L = 1\) → inconclusive. Best for: factorials (n!), exponentials (\(a^n\)), and combinations thereof. Critical simplifications: \(\frac{(n+1)!}{n!} = n+1\); \(\frac{a^{n+1}}{a^n} = a\). Always simplify BEFORE taking limit! Why it works: if ratio stays below 1, series behaves like convergent geometric series; if above 1, terms don't shrink to zero. Test gives absolute convergence, strongest form. Common failure: L = 1 for all p-series—use p-series test instead. Factorials grow faster than exponentials: \(\frac{a^n}{n!}\) always converges; \(\frac{n!}{a^n}\) always diverges (for constant a). This is THE power tool for complex series—appears on EVERY BC exam! Master simplification techniques! 🎯✨