Convergence Tests Summary

The Master Convergence Test Guide

Part 1 of 7 — Overview of All Tests

Every Convergence Test You Need

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 $|r|$
• $p$-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 $r^n$?

• Yes → Ratio Test

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

• Yes → Root Test

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.

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.

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):

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

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.

When to Use Which

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 ($p = 2$), $\sum (n+1)/(n^3-5)$ 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

LCT Limit

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.

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:

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

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 $u = \ln x$, $du = dx/x$:

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

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.

Absolute vs. Conditional Convergence

Part 4 of 7 — Three Categories

Convergence Classification

Every series falls into exactly one category:

The Hierarchy

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

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

Testing Procedure

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, $p = 3 > 1$ → 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 | |--------|-----------|----------|---------------| | $\sum (-1)^n/n^2$ | Conv ($p = 2$) | Conv | Absolute | | $\sum (-1)^n/n$ | Div (harmonic) | Conv (AST) | Conditional | | $\sum (-1)^n/\sqrt[3]{n}$ | Div ($p = 1/3$) | Conv (AST) | Conditional | | $\sum (-1)^n$ | Div | Div | Divergent |

Classification Practice

Classify These Series

Classification

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.

AP Exam Strategies — Convergence

Part 5 of 7 — Test Selection Under Pressure

AP Exam Convergence Problem Types

The 30-Second Flowchart for MC

$\boxed{\text{Divergence Test} \to \text{Geometric/p?} \to \text{Alternating?} \to \text{Ratio/Root?} \to \text{Comparison}}$

1. Quick check: $\lim a_n \neq 0$? → Diverges
2. Recognizable? Geometric or $p$-series → formula
3. Alternating? AST (verify $b_n \downarrow 0$)
4. Factorials or $n$th powers? Ratio or Root Test
5. 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

Common AP Pitfalls

AP Strategy Questions

Best Test Selection

Quick Decision

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.

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 $\sum a_n$

Next: Part 7 — Comprehensive Review.

Comprehensive Review — Convergence Tests

Part 7 of 7 — Final Assessment

Master Reference

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

Review Set A — Convergence/Divergence

Review Set B — Classification

Best Test Selection

Final Challenge

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

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