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Convergence Tests Summary

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Convergence Tests Summary - Complete Interactive Lesson

Part 1: Core Concepts

The Master Convergence Test Guide

Part 1 of 7 — Overview of All Tests

Every Convergence Test You Need

Test	Applies to	Conclusion
Divergence	Any $\sum a_n$	If $\lim a_n \neq 0$ , diverges
Geometric	$\sum ar^n$	$
$p$ -Series	$\sum 1/n^p$	$p > 1$ : converges
Integral	$f$ positive, decreasing	$\sum$ and $\int$ same behavior
Comparison	$0 \le a_n \le b_n$	$\sum b_n$ conv conv
Limit Comparison	$a_n, b_n > 0$	$\lim a_n/b_n = L > 0$ : same behavior
AST	$\sum (-1)^n b_n$	$b_n \downarrow 0$ : converges
Ratio	Any, best for $n!$	$L < 1$ : abs conv; $L > 1$ : div
Root	Any, best for $b_n^n$	Same thresholds as Ratio

The First Question: Does $a_n \to 0$ ?

$\boxed{\lim_{n \to \infty} a_n \neq 0 \quad \Longrightarrow \quad \sum a_n \text{ diverges}}$

Key Fact: Always check the Divergence Test first. It's free, fast, and catches many series immediately.

Decision Flowchart

Step 1: Is $\lim a_n \neq 0$ ? → Diverges (Divergence Test)

Step 2: Is it a known type?

Geometric: $\sum ar^n$ → check

Test Identification

Quick Test Selection

Divergence Test

Summary

9 convergence tests, each with specific strengths
Always start with Divergence Test, then look for known types
The flowchart guides you to the right test quickly
Divergence Test proves divergence only, never convergence

Next: Part 2 — Comparison Tests Deep Dive.

Part 2: Worked Examples

Comparison Tests Deep Dive

Part 2 of 7 — Direct & Limit Comparison

Direct Comparison Test (DCT)

For $0 \le a_n \le b_n$ for all $n$ (eventually):

Part 3: Problem-Solving Patterns

Integral Test & Special Series

Part 3 of 7 — When Other Tests Fail

The Integral Test

If $f(x)$ is positive, continuous, and decreasing for $x \ge N$ , and $a_n = f(n)$ , then:

Part 4: Graphs and Interpretation

Absolute vs. Conditional Convergence

Part 4 of 7 — Three Categories

Convergence Classification

Every series falls into exactly one category:

Category	Definition	Example
Absolutely convergent	$\sum	a_n
Conditionally convergent	$\sum a_n$ converges but $\sum	a_n
Divergent	$\sum a_n$ diverges

Part 5: Applications

AP Exam Strategies — Convergence

Part 5 of 7 — Test Selection Under Pressure

AP Exam Convergence Problem Types

Type	What to expect	Strategy
"Determine convergence"	Single series, pick a test	Use the flowchart
"Which test and why?"	Justify your choice	Name the test, verify hypotheses
"Determine interval of convergence"	Power series	Ratio Test for $R$ , test endpoints separately
"Absolute or conditional?"	Alternating series	Check $\sum
"FRQ series justification"	Part of larger problem	Be precise: state theorem, verify conditions

The 30-Second Flowchart for MC

Part 6: Exam Strategy

Problem-Solving Workshop

Part 6 of 7 — Mixed Practice

Work through these problems choosing the best test for each.

Workshop Set A — Identify & Classify

Workshop Set B — Test Selection

Classify Each Series

Compute the Limit

Workshop Complete

Key takeaways:

Always start with Divergence Test
Match the series structure to the best test
Classify: check $\sum |a_n|$ first, then $\sum a_n$

Part 7: Mixed Review

Comprehensive Review — Convergence Tests

Part 7 of 7 — Final Assessment

Master Reference

Test	Use when...	Conclusion
Divergence	Always first	$\lim a_n \neq 0 \Rightarrow$ diverges

\Rightarrow \sum a_n

p

-Series:

\sum 1/n^p

→ check

p

Telescoping:

\sum (b_n - b_{n+1})

→ evaluate

Step 3: Does it alternate?

Yes → AST (check $b_n$ decreasing, $\to 0$ )

Step 4: Contains factorials or $r^n$ ?

Step 5: Form $(b_n)^n$ ?

Step 6: Looks like a known series?

Yes → Limit Comparison or Direct Comparison

Step 7: Positive, decreasing, integratable?

Yes → Integral Test

AP Tip: This flowchart covers 95%+ of AP exam series. Memorize it.

$\sum b_n \text{ converges} \Rightarrow \sum a_n \text{ converges}$ $\sum a_n \text{ diverges} \Rightarrow \sum b_n \text{ diverges}$

Limit Comparison Test (LCT)

For $a_n, b_n > 0$ : if $\lim_{n \to \infty} a_n/b_n = L$ where $0 < L < \infty$ , then $\sum a_n$ and $\sum b_n$ have the same behavior.

Situation	Use
Easy to compare term-by-term	DCT
Hard to prove $a_n \le b_n$ directly	LCT
Series "looks like" a $p$ -series	LCT with $1/n^p$

Key Fact: LCT is usually easier on the AP exam because you don't need to prove an inequality — just compute a limit.

LCT Example: $\sum \frac{n+1}{n^3 - 5}$

Looks like $1/n^2$ for large $n$ . Compare with $b_n = 1/n^2$ :

$\lim \frac{(n+1)/(n^3-5)}{1/n^2} = \lim \frac{n^2(n+1)}{n^3 - 5} = \lim \frac{n^3 + n^2}{n^3 - 5} = 1$

Since $0 < 1 < \infty$ and $\sum 1/n^2$ converges ( $p = 2$ ), converges.

DCT Example: $\sum \frac{\sin^2 n}{n^2}$

$0 \le \sin^2 n \le 1$ , so $0 \le \sin^2 n/n^2 \le 1/n^2$ .

$\sum 1/n^2$ converges → $\sum \sin^2 n/n^2$ converges by DCT.

DCT Example: $\sum \frac{1}{n - \ln n}$

For large $n$ : $n - \ln n < n$ , so $1/(n - \ln n) > 1/n$ .

Since $\sum 1/n$ diverges and our series is term-by-term larger, $\sum 1/(n - \ln n)$ diverges by DCT.

AP Tip: For LCT, pick the comparison series by looking at the dominant terms in numerator and denominator.

Comparison Practice

Comparison Selection

Summary

DCT: prove $a_n \le b_n$ directly; used when comparison is obvious
LCT: compute $\lim a_n/b_n$ ; easier, more flexible
Pick comparison by looking at dominant terms
$L = 0$ or $L = \infty$ : partial results (one direction only)
$0 < L < \infty$ : both series have same behavior

Next: Part 3 — Integral Test and Unusual Series.

$\boxed{\sum_{n=N}^{\infty} a_n \text{ and } \int_N^{\infty} f(x)\,dx \text{ converge or diverge together}}$

When to Use the Integral Test

$a_n = 1/(n \ln n)$ → $\int dx/(x \ln x)$ , easy substitution
$a_n = 1/(n(\ln n)^2)$ → converges
$a_n = ne^{-n^2}$ → $\int xe^{-x^2}\,dx$ , substitution
Any series where you can anti-differentiate $f(x)$ easily

Important: The Test Does NOT Give the Sum

The integral gives the same convergence/divergence behavior, but:

$\sum a_n \neq \int f(x)\,dx$

The integral provides bounds, not the exact sum.

Key Fact: The Integral Test is the "test of last resort" for positive series that don't match other patterns. It's also how we PROVE the $p$ -Series Test.

Example 1: $\sum_{n=2}^{\infty} \frac{1}{n \ln n}$

$f(x) = 1/(x \ln x)$ : positive, decreasing for $x \ge 2$ .

$\int_2^{\infty} \frac{dx}{x \ln x}$ : let , :

$\int \frac{du}{u} = \ln u \Big|_{\ln 2}^{\infty} = \infty$

Diverges. So $\sum 1/(n \ln n)$ diverges.

Example 2: $\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}$

$\int_2^{\infty} \frac{dx}{x(\ln x)^2}: \quad \int \frac{du}{u^2} = -\frac{1}{u}\Big|_{\ln 2}^{\infty} = \frac{1}{\ln 2}$

Converges. So $\sum 1/(n(\ln n)^2)$ converges.

General Pattern

$\sum \frac{1}{n(\ln n)^p}: \quad \begin{cases} \text{converges} & p > 1 \\ \text{diverges} & p \le 1 \end{cases}$

This is like a "log- $p$ -series."

AP Tip: Integral Test problems on the AP exam usually involve $\ln n$ in the denominator where other tests fail.

Integral Test Practice

Integral Test Decisions

Integral Test Evaluation

Summary

Integral Test: same convergence behavior as the improper integral
Use when other tests fail, especially for series with $\ln n$
Log- $p$ -series: $\sum 1/(n(\ln n)^p)$ converges iff $p > 1$
The integral gives bounds, not the sum

Next: Part 4 — Absolute vs. Conditional Convergence.

$\boxed{\text{Absolute convergence} \Rightarrow \text{Convergence}}$

The converse is FALSE: convergence does NOT imply absolute convergence.

Step 1: Check $\sum |a_n|$ .

If it converges → absolutely convergent (done!)

Step 2: If $\sum |a_n|$ diverges, check $\sum a_n$ .

If $\sum a_n$ converges (usually by AST) → conditionally convergent
If $\sum a_n$ diverges → divergent

Key Fact: "Absolute convergence" means you can rearrange the terms in any order and still get the same sum. Conditionally convergent series can be rearranged to sum to ANY value (Riemann's rearrangement theorem).

Example 1: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}$

$\sum |a_n| = \sum 1/n^3$ . $p$ -Series, → converges.

Absolutely convergent.

Example 2: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ (alternating harmonic)

$\sum |a_n| = \sum 1/n$ → diverges (harmonic).

$\sum a_n = \sum (-1)^{n+1}/n$ → converges by AST.

Conditionally convergent.

Example 3: $\sum_{n=1}^{\infty} \frac{(-1)^n n}{n+1}$

$\lim |a_n| = \lim n/(n+1) = 1 \neq 0$ .

Divergent (by Divergence Test — doesn't even converge).

Quick Classification Guide

| Series | $\sum|a_n|$ | $\sum a_n$ | Classification | |--------|-----------|----------|---------------| | $\sum (- 1)^{n}$ | Conv () | Conv | Absolute | | | Div (harmonic) | Conv (AST) | Conditional | | | Div () | Conv (AST) | Conditional | | | Div | Div | Divergent |

Classification Practice

Classify These Series

Summary

Three categories: absolute, conditional, divergent
Check $\sum |a_n|$ first; if it converges, done (absolute)
If $\sum |a_n|$ diverges but $\sum a_n$ converges → conditional
Absolute ⇒ convergent, but not vice versa

Next: Part 5 — The Hardest AP Problems.

Divergence Test \to Geometric/p? \to Alternating? \to Ratio/Root? \to Comparison

Quick check: $\lim a_n \neq 0$ ? → Diverges
Recognizable? Geometric or $p$ -series → formula
Alternating? AST (verify $b_n \downarrow 0$ )
Factorials or $n$ th powers? Ratio or Root Test
Compare: DCT or LCT with known series

AP Tip: On FRQs, always state the test name, verify ALL conditions, and write a concluding statement.

Writing Perfect Justifications

Bad answer (no credit): "It converges by comparison."

Good answer (full credit): "Since $0 \leq \frac{1}{n^2 + 1} \leq \frac{1}{n^2}$ for all $n \geq 1$ , and $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges ( $p$ -series, $p = 2 > 1$ ), by the Direct Comparison Test, $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$ converges."

FRQ Checklist

Step	Example
State the test	"By the Ratio Test..."
Verify conditions	"Since $a_n > 0$ and $\lim_{n\to\infty} a_{n+1}/a_n = L$ ..."

Common AP Pitfalls

Mistake	Why it loses points
Not checking $\lim a_n = 0$ first	Divergence Test is always first
Saying "converges by Divergence Test"	Div Test can only prove divergence
Forgetting endpoint checks for IOC	$R$ alone is not the full answer
LCT: not choosing the right comparison	Compare to , not

AP Strategy Questions

Best Test Selection

Summary

Use the flowchart: Div Test → recognize → alternating → ratio/root → comparison
FRQs: name the test, verify conditions, write a conclusion
Common errors: "converges by Divergence Test," forgetting endpoint tests, weak comparisons

Next: Part 6 — Problem-Solving Workshop.

Next: Part 7 — Comprehensive Review.

$\boxed{\text{Flowchart: Div Test} \to \text{Recognizable} \to \text{Alternating} \to \text{Ratio/Root} \to \text{Comparison}}$

Review Set A — Convergence/Divergence

Review Set B — Classification

Best Test Selection

Convergence Tests Summary — Complete

You've mastered:

All 9 convergence tests and when to use each
Direct and Limit Comparison Tests
Integral Test and special series (log- $p$ )
Absolute vs. conditional convergence
AP exam strategy and justification writing

$\boxed{\text{Master the flowchart. Verify all conditions. Write clear conclusions.}}$

\sum (n+1)/(n^3-5)

{convergesdivergesp>1p≤1

\sum (- 1)^{n} / n^{2}

\sum (-1)^n/\sqrt[3]{n}

Convergence Tests Summary

Limit Comparison Test (LCT)

When to Use Which

LCT Example: $\sum \frac{n+1}{n^3 - 5}$

DCT Example: $\sum \frac{\sin^2 n}{n^2}$

DCT Example: $\sum \frac{1}{n - \ln n}$

Summary

When to Use the Integral Test

Important: The Test Does NOT Give the Sum

Example 1: $\sum_{n=2}^{\infty} \frac{1}{n \ln n}$

Example 2: $\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}$

General Pattern

Summary

The Hierarchy

Testing Procedure

Example 1: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}$

Example 2: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ (alternating harmonic)

Example 3: $\sum_{n=1}^{\infty} \frac{(-1)^n n}{n+1}$

Quick Classification Guide

Summary

Writing Perfect Justifications

FRQ Checklist

Common AP Pitfalls

Summary

Convergence Tests Summary — Complete

Convergence Tests Summary

Convergence Tests Summary - Complete Interactive Lesson

Part 1: Core Concepts

The Master Convergence Test Guide

Every Convergence Test You Need

The First Question: Does an→0a_n \to 0an​→0?

Decision Flowchart

Summary

Part 2: Worked Examples

Comparison Tests Deep Dive

Direct Comparison Test (DCT)

Part 3: Problem-Solving Patterns

Integral Test & Special Series

The Integral Test

Part 4: Graphs and Interpretation

Absolute vs. Conditional Convergence

Convergence Classification

Part 5: Applications

AP Exam Strategies — Convergence

AP Exam Convergence Problem Types

The 30-Second Flowchart for MC

Part 6: Exam Strategy

Problem-Solving Workshop

Workshop Complete

Part 7: Mixed Review

Comprehensive Review — Convergence Tests

Master Reference

Limit Comparison Test (LCT)

When to Use Which

LCT Example: ∑n+1n3−5\sum \frac{n+1}{n^3 - 5}∑n3−5n+1​

DCT Example: ∑sin⁡2nn2\sum \frac{\sin^2 n}{n^2}∑n2sin2n​

DCT Example: ∑1n−ln⁡n\sum \frac{1}{n - \ln n}∑n−lnn1​

Summary

When to Use the Integral Test

Important: The Test Does NOT Give the Sum

Example 1: ∑n=2∞1nln⁡n\sum_{n=2}^{\infty} \frac{1}{n \ln n}∑n=2∞​nlnn1​

Example 2: ∑n=2∞1n(ln⁡n)2\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}∑n=2∞​n(lnn)

General Pattern

Summary

The Hierarchy

Testing Procedure

Example 1: ∑n=1∞(−1)nn3\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}∑n=1∞​n3(−1)n​

Example 2: ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}∑n=1∞​n(−1) (alternating harmonic)

Example 3: ∑n=1∞(−1)nnn+1\sum_{n=1}^{\infty} \frac{(-1)^n n}{n+1}∑n=1∞​n+1(−1

Quick Classification Guide

Summary

Writing Perfect Justifications

FRQ Checklist

Common AP Pitfalls

Summary

Convergence Tests Summary — Complete

The First Question: Does $a_n \to 0$ ?

LCT Example: $\sum \frac{n+1}{n^3 - 5}$

DCT Example: $\sum \frac{\sin^2 n}{n^2}$

DCT Example: $\sum \frac{1}{n - \ln n}$

Example 1: $\sum_{n=2}^{\infty} \frac{1}{n \ln n}$

Example 2: $\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2}$

Example 1: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}$

Example 2: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ (alternating harmonic)

Example 3: $\sum_{n=1}^{\infty} \frac{(-1)^n n}{n+1}$