Infinite Series

Infinite Series — Convergence Tests

Part 1 of 7 — The Integral Test

Review: Series Convergence

$\sum_{n=1}^\infty a_n$ converges if and only if $\lim_{N \to \infty} S_N$ exists (finite).

We already know:

• Geometric series: $|r| < 1$
• $p$-series: $p > 1$
• $n$th term test: $a_n \not\to 0 \implies$ divergence

Now we develop systematic convergence tests.

The Integral Test

If $f(x)$ is continuous, positive, and decreasing for $x \ge 1$, and $a_n = f(n)$, then:

$\boxed{\sum_{n=1}^\infty a_n \text{ and } \int_1^\infty f(x)\,dx \text{ either both converge or both diverge.}}$

Key Fact: The integral test does NOT give the sum — only whether the series converges.

Example 1 — Proving $p$-series

Show $\sum 1/n^2$ converges using the integral test.

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

$\int_1^\infty \frac{1}{x^2}\,dx = \lim_{b \to \infty}\left[-\frac{1}{x}\right]_1^b = 0 - (-1) = 1$

Integral converges $\implies$ series converges. ✓

Example 2 — Harmonic series diverges

$f(x) = 1/x$: continuous, positive, decreasing.

$\int_1^\infty \frac{1}{x}\,dx = \lim_{b \to \infty} \ln b = \infty$

Integral diverges $\implies$ $\sum 1/n$ diverges. ✓

Remainder Estimate

If $\sum a_n$ converges and $S$ denotes its sum, then:

$\int_{N+1}^\infty f(x)\,dx \le S - S_N \le \int_N^\infty f(x)\,dx$

Practice Problems

Concept Checks

Computation

Summary

• Integral test: compare $\sum a_n$ with $\int f(x)\,dx$ (same convergence behavior)
• Requires: continuous, positive, decreasing $f$
• Does NOT give the sum, only convergence/divergence
• Useful for proving $p$-series results and testing unfamiliar series

Next: Part 2 — Comparison and Limit Comparison tests.

Infinite Series — Comparison Tests

Part 2 of 7 — Direct & Limit Comparison Tests

Direct Comparison Test (DCT)

For $0 \le a_n \le b_n$:

Intuition: Smaller than convergent $\implies$ convergent. Bigger than divergent $\implies$ divergent.

Limit Comparison Test (LCT)

If $a_n, b_n > 0$ and $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ where $0 < L < \infty$, then:

$\boxed{\sum a_n \text{ and } \sum b_n \text{ either both converge or both diverge.}}$

AP Tip: The LCT is the most versatile comparison test. Choose $b_n$ to be a simpler series ($p$-series or geometric) that behaves like $a_n$.

Examples

Example 1 (DCT): $\sum \frac{1}{n^2 + n}$

$\frac{1}{n^2 + n} < \frac{1}{n^2}$ and $\sum 1/n^2$ converges ($p = 2$).

By DCT, $\sum \frac{1}{n^2 + n}$ converges. ✓

Example 2 (LCT): $\sum \frac{3n^2 + 1}{n^4 - 2n + 7}$

Compare with $b_n = 1/n^2$ (dominant terms give $3n^2/n^4 = 3/n^2$).

$\frac{a_n}{b_n} = \frac{(3n^2+1)n^2}{n^4-2n+7} \to \frac{3n^4}{n^4} = 3$

Since $L = 3 \in (0, \infty)$ and $\sum 1/n^2$ converges, the given series converges by LCT. ✓

Example 3 (LCT): $\sum \frac{1}{\sqrt{n} - 1}$

Compare with $b_n = 1/\sqrt{n}$: $\frac{a_n}{b_n} = \frac{\sqrt{n}}{\sqrt{n}-1} \to 1$.

$\sum 1/\sqrt{n}$ diverges ($p = 1/2$), so $\sum \frac{1}{\sqrt{n}-1}$ diverges. ✓

Practice

Test Selection

LCT Practice

Summary

• DCT: Bound $a_n$ above by convergent or below by divergent
• LCT: Compare $a_n/b_n \to L \in (0, \infty)$ — same behavior
• Choose $b_n$ by identifying dominant terms
• Both tests require positive terms

Next: Part 3 — The Ratio and Root Tests.

Infinite Series — Ratio & Root Tests

Part 3 of 7 — The Ratio and Root Tests

The Ratio Test

Let $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$. Then:

The Root Test

Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. Same conclusions as ratio test.

Key Fact: The ratio test works best with factorials and exponentials. The root test works best with $n$th powers. Both are inconclusive for $p$-series.

Examples

Ratio Test: $\sum \frac{n!}{3^n}$

$\frac{a_{n+1}}{a_n} = \frac{(n+1)!}{3^{n+1}} \cdot \frac{3^n}{n!} = \frac{n+1}{3} \to \infty$

$L = \infty > 1$: diverges.

Ratio Test: $\sum \frac{2^n}{n!}$

$\frac{a_{n+1}}{a_n} = \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} = \frac{2}{n+1} \to 0$

$L = 0 < 1$: converges absolutely.

Root Test: $\sum \left(\frac{n}{2n+1}\right)^n$

$\sqrt[n]{a_n} = \frac{n}{2n+1} \to \frac{1}{2}$

$L = 1/2 < 1$: converges absolutely.

Practice Problems

Test Selection

Ratio Test Computation

Summary

• Ratio test: $L = \lim |a_{n+1}/a_n|$ — best for factorials and exponentials
• Root test: $L = \lim \sqrt[n]{|a_n|}$ — best for $n$th powers
• $L < 1$: converges; $L > 1$: diverges; $L = 1$: inconclusive
• Both tests are inconclusive for $p$-series (use comparison or integral test instead)

$\boxed{L < 1 \implies \text{converges} \qquad L > 1 \implies \text{diverges} \qquad L = 1 \implies \text{inconclusive}}$

Next: Part 4 — Absolute and conditional convergence.

Infinite Series — Absolute & Conditional Convergence

Part 4 of 7 — Types of Convergence

Definitions

Key Theorem

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

But NOT vice versa!

The Classic Example

$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2$

This converges, but $\sum 1/n$ diverges, so it converges conditionally.

AP Tip: When a problem asks "does the series converge absolutely, conditionally, or diverge?" — test $\sum |a_n|$ first. If it converges, you're done (absolute). If not, check $\sum a_n$ separately.

Strategy for Classification

Step 1: Test $\sum |a_n|$ for convergence.

• If $\sum |a_n|$ converges → absolutely convergent ✓

Step 2: If $\sum |a_n|$ diverges, test $\sum a_n$ (typically with AST).

• If $\sum a_n$ converges → conditionally convergent
• If $\sum a_n$ diverges → divergent

Why Conditional Convergence Matters

Conditionally convergent series have surprising properties:

• Riemann Rearrangement Theorem: By rearranging terms, you can make the series sum to ANY value (or diverge). This is why absolute convergence is "safer."
• On the AP exam, conditionally convergent series typically appear in the context of interval of convergence endpoints.

Practice Problems

Classification Practice

Quick Classification

Summary

• Absolute convergence: $\sum |a_n|$ converges
• Conditional convergence: $\sum a_n$ converges but $\sum |a_n|$ diverges
• Absolute convergence $\implies$ convergence (not vice versa)
• Test $\sum |a_n|$ first, then $\sum a_n$ if needed

Next: Part 5 — Choosing the right test (decision flowchart).

Infinite Series — Choosing the Right Test

Part 5 of 7 — Convergence Test Strategy

Decision Flowchart for $\sum a_n$

$\boxed{\text{No single test works for all series — practice builds intuition.}}$

AP Tip: The exam frequently asks "which test is appropriate?" or "justify your answer using [a specific test]." Know the hypotheses of each test cold.

Test Selection Examples

Example 1: $\sum_{n=1}^\infty \frac{3^n}{n!}$

• Contains $n!$ → Ratio Test: $\frac{a_{n+1}}{a_n} = \frac{3}{n+1} \to 0 < 1$ → converges ✓

Example 2: $\sum_{n=1}^\infty \frac{n^2}{3n^2 + 1}$

• $a_n \to 1/3 \neq 0$ → Divergence Test → diverges ✓

Example 3: $\sum_{n=2}^\infty \frac{1}{n \ln n}$

• Positive, decreasing, integratable → Integral Test: $\int_2^\infty \frac{dx}{x\ln x} = \ln(\ln x)\big|_2^\infty = \infty$ → diverges ✓

Example 4: $\sum_{n=1}^\infty \frac{(-1)^n n}{n+1}$

• $|a_n| = n/(n+1) \to 1 \neq 0$ → Divergence Test → diverges ✓ (not alternating — fails hypothesis)

Common Pitfalls

Which Test? Practice

Test Strategy Application

Ratio Test Application

Key Takeaways

Key Fact: On the AP exam, you'll almost never need more than one test per series. The challenge is identifying which one.

Next: Part 6 — Problem-Solving Workshop.

Infinite Series — Problem-Solving Workshop

Part 6 of 7 — Practice with All Tests

Work through these problems carefully. For each series, identify the appropriate convergence test, verify hypotheses, and state a conclusion.

Warm-Up: Test Identification

For each series, think about which test best applies before solving.

Workshop Problems

Classify Each Series

Computation Challenge

Workshop Takeaways

• Always check $a_n \to 0$ first (Divergence Test)
• Factorials and exponentials → Ratio Test
• Powers of $n$ → Comparison / $p$-series
• Alternating signs → AST (after verifying $b_n$ is decreasing and $\to 0$)
• For classification: test $\sum |a_n|$ first, then $\sum a_n$

Next: Part 7 — Comprehensive Review.

Infinite Series — Comprehensive Review

Part 7 of 7 — Review All Convergence Tests

Quick Reference: All Tests

$\boxed{\text{Absolute convergence} \implies \text{Convergence} \implies a_n \to 0}$

AP Exam Note: You MUST state the test name and verify its hypotheses for full credit. A correct answer with no justification earns minimal credit.

Comprehensive MC Review

More Review Problems

Final Classification Drill

Final Computation

Infinite Series — Complete Summary

You've mastered:

• Integral Test — connects series and improper integrals
• Comparison Tests — DCT and LCT for bounding series
• Ratio & Root Tests — best for factorials, exponentials, and $n$-th powers
• Absolute vs. Conditional Convergence — fundamental classification
• Test Selection Strategy — choosing the right tool for each series

Key Fact: Series convergence is a major BC topic, typically appearing in both MC and FRQ sections. Expect 3-5 questions on the AP exam.

Up Next: Alternating Series — deep dive into the Alternating Series Test, error bounds, and applications.