🎯⭐ INTERACTIVE LESSON

Infinite Series

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Infinite Series - Complete Interactive Lesson

Part 1: Partial Sums & Geometric 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: $a_{n} \to̸ 0$ 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 . ✓

Practice Problems

Concept Checks

Computation

Summary

Integral test: compare $\sum a_n$ with $\int f(x)\,dx$ (same convergence behavior)
Requires: continuous, positive, decreasing $f$

Part 2: Telescoping Series & Divergence Test

Infinite Series — Comparison Tests

Part 2 of 7 — Direct & Limit Comparison Tests

Direct Comparison Test (DCT)

For $0 \le a_n \le b_n$ :

If...	Then...

Part 3: Integral Test & p-Series

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:

Part 4: Comparison Tests

Infinite Series — Absolute & Conditional Convergence

Part 4 of 7 — Types of Convergence

Definitions

Type	Definition
Absolutely convergent	$\sum
Conditionally convergent	$\sum a_n$ converges but $\sum
Divergent	$\sum a_n$ does not converge

Part 5: Ratio & Root Tests

Infinite Series — Choosing the Right Test

Part 5 of 7 — Convergence Test Strategy

Decision Flowchart for $\sum a_n$

Step	Ask Yourself	Action
1	Does $a_n \to 0$ ?

Part 6: Practice 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.

Series	Key Feature	Best Test
$\sum n!/n^n$	Both $n!$ and

Part 7: Final Assessment

Infinite Series — Comprehensive Review

Part 7 of 7 — Review All Convergence Tests

Quick Reference: All Tests

Test	Hypotheses	Conclusion
Divergence	$\lim a_n \neq 0$	Diverges

a_{n} \neq \to 0 ⟹

either both converge or both diverge.

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

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. ✓

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

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

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.

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 $b_n = 1/\sqrt{n}$ : $\frac{a_{n}}{}$ .

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

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.

$L$ value	Conclusion
$L < 1$	Converges absolutely
$L > 1$ (or $L = \infty$ )	Diverges
$L = 1$	Inconclusive

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

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.

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

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

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).

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

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

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

Common Pitfalls

Mistake	Correction
AST on $	a_n
Ratio/Root gives $L = 1$	Test is inconclusive — try another
Comparison in wrong direction	$a_n \le b_n$ and converges → converges. NOT the other way for convergence

Which Test? Practice

Test Strategy Application

Ratio Test Application

Key Takeaways

Situation	Go-To Test
$a_n \not\to 0$	Divergence Test
Factorials or exponentials	Ratio Test
$n$ -th power structure	Root Test
Polynomial-like terms	Comparison / LCT
Positive, continuous, decreasing	Integral Test
Alternating signs	AST

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.

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.

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

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.

n4−2n+7(3n2+1)n2→

\frac{a _{n}}{b _{n}} = \frac{n}{n - 1} \to 1

\sum \frac{1}{n - 1}

Infinite Series

Example 2 — Harmonic series diverges

Remainder Estimate

Limit Comparison Test (LCT)

Examples

Summary

The Root Test

Examples

Summary

Key Theorem

The Classic Example

Strategy for Classification

Why Conditional Convergence Matters

Summary

Test Selection Examples

Common Pitfalls

Key Takeaways

Workshop Takeaways

Infinite Series — Complete Summary

$\sum b_n$ converges	$\sum a_n$ converges
$\sum a_n$ diverges	$\sum b_n$ diverges

Infinite Series

Infinite Series - Complete Interactive Lesson

Part 1: Partial Sums & Geometric Series

Infinite Series — Convergence Tests

Review: Series Convergence

The Integral Test

Example 1 — Proving ppp-series

Summary

Part 2: Telescoping Series & Divergence Test

Infinite Series — Comparison Tests

Direct Comparison Test (DCT)

Part 3: Integral Test & p-Series

Infinite Series — Ratio & Root Tests

The Ratio Test

Part 4: Comparison Tests

Infinite Series — Absolute & Conditional Convergence

Definitions

Part 5: Ratio & Root Tests

Infinite Series — Choosing the Right Test

Decision Flowchart for ∑an\sum a_n∑an​

Part 6: Practice Workshop

Infinite Series — Problem-Solving Workshop

Warm-Up: Test Identification

Part 7: Final Assessment

Infinite Series — Comprehensive Review

Quick Reference: All Tests

Example 2 — Harmonic series diverges

Remainder Estimate

Limit Comparison Test (LCT)

Examples

Summary

The Root Test

Examples

Summary

Key Theorem

The Classic Example

Strategy for Classification

Why Conditional Convergence Matters

Summary

Test Selection Examples

Common Pitfalls

Key Takeaways

Workshop Takeaways

Infinite Series — Complete Summary

Example 1 — Proving $p$ -series

Decision Flowchart for $\sum a_n$