Unit 10.7 – Alternating Series Test for Convergence BC ONLY

AP® Calculus BC | Testing Series with Alternating Signs

Why This Matters: The Alternating Series Test (AST) handles series where terms alternate between positive and negative! Many important series (like the alternating harmonic series) converge even though their absolute versions diverge. Plus, this test gives us ERROR BOUNDS—you can estimate how close partial sums are to the true sum!

🎯 What is an Alternating Series?

ALTERNATING SERIES DEFINITION

Form 1: Starting with Positive

\[ \sum_{n=1}^{\infty} (-1)^{n+1} b_n = b_1 - b_2 + b_3 - b_4 + b_5 - \cdots \]

where \(b_n > 0\) for all n

Form 2: Starting with Negative

\[ \sum_{n=1}^{\infty} (-1)^n b_n = -b_1 + b_2 - b_3 + b_4 - b_5 + \cdots \]

where \(b_n > 0\) for all n

📝 Key Point: The \(b_n\) terms are all POSITIVE. The alternating sign comes from \((-1)^n\) or \((-1)^{n+1}\).

⭐ The Alternating Series Test

Alternating Series Test (AST)

THE TEST:

For the alternating series \(\sum (-1)^{n+1} b_n\) or \(\sum (-1)^n b_n\),

if the following THREE conditions hold:

Condition 1: Positive Terms

\[ b_n > 0 \text{ for all } n \]

Condition 2: Decreasing

\[ b_{n+1} \leq b_n \text{ for all } n \geq N \text{ (some N)} \]

Terms are decreasing in magnitude

Condition 3: Limit to Zero

\[ \lim_{n \to \infty} b_n = 0 \]

THEN the series CONVERGES!

📏 Alternating Series Error Bound

The Error Bound (SUPER IMPORTANT!)

ERROR ESTIMATE:

If \(\sum (-1)^{n+1} b_n\) converges to \(S\) and \(S_n\) is the n-th partial sum, then:

\[ |S - S_n| \leq b_{n+1} \]

The error is at most the FIRST OMITTED TERM!

What This Means:

If you stop at \(S_n\) (sum of first n terms), error ≤ \(b_{n+1}\)
To guarantee error < 0.001, find n where \(b_{n+1} < 0.001\)
This ONLY works for alternating series that pass the AST!

📝 Visual: Partial sums bounce back and forth, getting closer to S. Each bounce overshoots by at most the next term!

🔄 Absolute vs. Conditional Convergence

IMPORTANT DISTINCTION

Absolute Convergence:

\(\sum a_n\) converges absolutely if \(\sum |a_n|\) converges

If series converges absolutely, it also converges (regular convergence)

Conditional Convergence:

\(\sum a_n\) converges conditionally if:

\(\sum a_n\) converges, BUT
\(\sum |a_n|\) diverges

⚠️ KEY FACT: Absolute convergence is STRONGER than conditional convergence!

\[ \text{Absolute convergence} \Rightarrow \text{Convergence (but not vice versa)} \]

📖 Comprehensive Worked Examples

Example 1: Classic Alternating Harmonic Series

Problem: Does \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots\) converge?

Solution:

Check AST conditions with \(b_n = \frac{1}{n}\):

Condition 1: Positive?

\(b_n = \frac{1}{n} > 0\) ✓

Condition 2: Decreasing?

\(\frac{1}{n+1} < \frac{1}{n}\) for all \(n \geq 1\) ✓

Condition 3: Limit to zero?

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

All conditions satisfied → Series CONVERGES by AST

(Actually converges to ln 2!)

Note: Regular harmonic series \(\sum \frac{1}{n}\) DIVERGES, but alternating version CONVERGES! This is conditional convergence.

Example 2: Error Bound Application

Problem: How many terms of \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}\) are needed to approximate the sum within 0.01?

First verify it converges:

\(b_n = \frac{1}{n^2}\): positive ✓, decreasing ✓, limit = 0 ✓

Converges by AST

Use error bound:

Need \(|S - S_n| \leq b_{n+1} < 0.01\)

\[ \frac{1}{(n+1)^2} < 0.01 \]

\[ (n+1)^2 > 100 \]

\[ n+1 > 10 \quad \Rightarrow \quad n \geq 10 \]

ANSWER: Need 10 terms

Example 3: Checking Decreasing

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

Let \(b_n = \frac{n}{n^2+1}\)

Condition 1: \(b_n > 0\) ✓

Condition 2: Check decreasing

Method: Show \(b_{n+1} < b_n\) or use derivative

Let \(f(x) = \frac{x}{x^2+1}\), find \(f'(x)\):

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

For \(x \geq 1\): \(f'(x) < 0\), so decreasing ✓

Condition 3: Limit

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

Series CONVERGES by AST

Example 4: Absolute vs. Conditional Convergence

Problem: Determine if \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\) converges absolutely or conditionally.

Check absolute convergence:

\[ \sum_{n=1}^{\infty} \left|\frac{(-1)^{n+1}}{n}\right| = \sum_{n=1}^{\infty} \frac{1}{n} \]

This is the harmonic series, which DIVERGES

Check regular convergence:

From Example 1, we know this converges by AST

Series converges but NOT absolutely → CONDITIONALLY CONVERGENT

Example 5: Absolute Convergence

Problem: Does \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}\) converge absolutely or conditionally?

Check absolute convergence:

\[ \sum_{n=1}^{\infty} \left|\frac{(-1)^n}{n^2}\right| = \sum_{n=1}^{\infty} \frac{1}{n^2} \]

This is p-series with p=2 > 1, so CONVERGES

Series converges ABSOLUTELY (which implies regular convergence too)

📊 Quick Reference Table

Common Alternating Series
Series	Converges?	Type
\(\sum \frac{(-1)^{n+1}}{n}\)	YES (AST)	Conditional
\(\sum \frac{(-1)^n}{n^2}\)	YES	Absolute
\(\sum \frac{(-1)^{n+1}}{\sqrt{n}}\)	YES (AST)	Conditional
\(\sum \frac{(-1)^n}{n^{3/2}}\)	YES	Absolute
\(\sum (-1)^n\)	NO	Diverges (limit ≠ 0)

💡 Essential Tips & Strategies

✅ Success Strategies:

Look for \((-1)^n\) or \((-1)^{n+1}\): Sign of alternating series
Check all THREE conditions: Positive, decreasing, limit = 0
Use derivative to check decreasing: Often easiest method
Error bound is powerful: Gives actual numerical accuracy
Check absolute convergence first: If \(\sum |a_n|\) converges, done!
Conditional convergence is weaker: Series rearrangements can change sum!
State "by AST": Always name the test on exams
Verify limit = 0: If not, series diverges by nth-term test

🔥 Quick Checklist for AST:

Identify alternating pattern \((-1)^n\) or \((-1)^{n+1}\)
Extract \(b_n\) (positive terms)
Check: \(b_n > 0\)?
Check: \(b_n\) decreasing? (use derivative or direct comparison)
Check: \(\lim b_n = 0\)?
If all yes → converges by AST
For error: first omitted term = \(b_{n+1}\)

❌ Common Mistakes to Avoid

Mistake 1: Forgetting to check if terms are decreasing
Mistake 2: Not verifying \(\lim b_n = 0\) (can't assume!)
Mistake 3: Using error bound when series doesn't satisfy AST
Mistake 4: Confusing \(b_{n+1}\) (error) with \(b_n\) (last term included)
Mistake 5: Thinking conditional convergence = divergence
Mistake 6: Not checking absolute convergence first
Mistake 7: Applying AST to non-alternating series
Mistake 8: Wrong sign pattern (mixing up \((-1)^n\) vs \((-1)^{n+1}\))
Mistake 9: Not showing decreasing condition on exam
Mistake 10: Forgetting AST gives convergence, not the sum value

📝 Practice Problems

Determine convergence and type (if applicable):

\(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^3}\)
\(\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\)
\(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n+1}\)
\(\sum_{n=2}^{\infty} \frac{(-1)^n}{\ln n}\)
How many terms of \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n!}\) for error < 0.001?

Answers:

Converges ABSOLUTELY (p-series p=3)
Converges CONDITIONALLY (AST works, but \(\sum \frac{1}{\sqrt{n}}\) diverges)
DIVERGES (limit = 1 ≠ 0, fails nth-term test)
Converges by AST (positive, decreasing, limit = 0)
Need 7 terms (since \(\frac{1}{8!} < 0.001\))

✏️ AP® Exam Success Tips

What AP® Graders Look For:

State "Alternating Series Test": Name the test explicitly
Verify all three conditions: Show positive, decreasing, limit = 0
For decreasing: Use derivative or algebraic comparison
Show limit calculation: \(\lim_{n \to \infty} b_n = 0\)
For error bounds: Identify \(b_{n+1}\) and solve inequality
Distinguish absolute/conditional: Check \(\sum |a_n|\) separately
State conclusion clearly: "Converges by AST" or "Conditionally convergent"
Show work for each condition: Don't just state "satisfied"

💯 Exam Strategy:

Check if series is alternating (look for \((-1)^n\))
Extract \(b_n\) (positive terms)
Verify \(b_n > 0\)
Check if \(b_n\) is decreasing (derivative method fastest)
Calculate \(\lim_{n \to \infty} b_n\)
If all conditions met, conclude convergence by AST
If asked about absolute/conditional, test \(\sum |a_n|\)
For error problems, use \(|error| \leq b_{n+1}\)

⚡ Quick Reference Guide

ALTERNATING SERIES TEST ESSENTIALS

The Test (THREE conditions):

\(b_n > 0\) (positive terms)
\(b_{n+1} \leq b_n\) (decreasing)
\(\lim_{n \to \infty} b_n = 0\)

All three → CONVERGES

Error Bound:

\[ |S - S_n| \leq b_{n+1} \]

Error ≤ first omitted term!

Convergence Types:

Absolute: \(\sum |a_n|\) converges
Conditional: \(\sum a_n\) converges but \(\sum |a_n|\) diverges

Remember:

Check ALL three conditions!
Error bound only for AST series!
Absolute → Convergence (not reverse)!

Master the Alternating Series Test! The AST requires three conditions: (1) \(b_n > 0\), (2) \(b_n\) decreasing, (3) \(\lim b_n = 0\). If all satisfied, series \(\sum (-1)^n b_n\) or \(\sum (-1)^{n+1} b_n\) converges. Error bound: \(|S - S_n| \leq b_{n+1}\)—error is at most the first omitted term! This is UNIQUE to alternating series. Absolute convergence: if \(\sum |a_n|\) converges, series converges absolutely (strongest). Conditional convergence: \(\sum a_n\) converges but \(\sum |a_n|\) diverges (weaker). Classic example: alternating harmonic \(\sum \frac{(-1)^{n+1}}{n}\) converges conditionally (to ln 2) while \(\sum \frac{1}{n}\) diverges. Always check absolute first—if that converges, done! AST is powerful for series where cancellation helps. Appears frequently on BC exams—master error bound calculations! 🎯✨