The Divergence Test

Lesson 01: The First Line of Defense

Logic Lab Infinite Series Analysis

The Harmonic Paradox

Consider the following infinite sum:

\[ \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \]

Notice the individual terms:

1, 0.5, 0.33, 0.25, 0.2, ... They are getting smaller and smaller!

Lab Poll

If the terms eventually approach zero, does the total sum stay finite or grow to infinity?

A

It stays finite.

B

It grows to infinity.

The n-th Term Test

Logic Rule

If \[ \lim_{n \to \infty} a_n \neq 0 \] or the limit does not exist, then the series \[ \sum_{n=1}^{\infty} a_n \] diverges.

Warning

The converse is NOT necessarily true. If the limit is zero, the test is inconclusive.

Strategy

Always use this test first! It is the fastest way to prove divergence.

Logic Case 01

Analyze the following:

\[ \sum_{n=1}^{\infty} \frac{n^2 + 1}{3n^2 - 5} \]

Identify the n-th term \( a_n \).
Evaluate \( \lim_{n \to \infty} a_n \).
Draw your conclusion.

Calculation

\[ \lim_{n \to \infty} \frac{n^2+1}{3n^2-5} = \frac{1}{3} \]

Conclusion

Since \( \frac{1}{3} \neq 0 \), the series diverges.

Necessity vs Sufficiency

Necessity

Terms approaching zero is necessary for convergence.

"If you want to live in the house of convergence, you MUST first walk through the door of zero."

Sufficiency

Terms approaching zero is not sufficient for convergence.

"Just because you are at the door does not mean you are inside the house."

Example: The Harmonic Series (\( 1/n \to 0 \)) but diverges!

Lab Protocol

1

Check the limit of the n-th term first.

2

If the limit is 0, the test is a "FAIL" (inconclusive). Pick a different tool.

3

If the limit is anything else (or DNE), you're done: Divergence Confirmed.

LOGIC LAB // MODULE 01 READY FOR FIELD WORK

Infinite Series Logic Lab

The Divergence Test

The Integral Test

Comparison Tests

The Ratio Test

Strategic Series Analysis

Divergence Test Slides

Divergence Test Teacher Guide

Harmonic Paradox Worksheet

Integral Test Slides

Integral Test Teacher Guide

Area Bound Worksheet

Comparison Test Slides

Comparison Test Teacher Guide

Benchmark Battle Worksheet

Ratio Test Slides

Ratio Test Teacher Guide

Growth Race Worksheet

Strategic Analysis Slides

Series Speed Dating Activity

Strategic Analysis Teacher Guide

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Jessica Waters

Statistiques Exploratoires

Hugo Pierrard

Radical Relationships

Calculus Foundations

Urban Concurrency Planning

Transformation Surgeons

Power Root Mastery

Sigma Summit

Power of Growth

Pattern Detectives

Recursive Sequence Mastery

Exponent Mastery

Slope and Symmetry

Algebra Rescue Polynomials

Domain Restriction Diagnostics

Optimal Apex Optimization

Slope Mastery

Polynomial Powers

Exponential Strategy Masterclass