Series Sleuths

Mastering the mechanics of geometric sequences, series, and the critical check for convergence.

12th Grade / Undergraduate Mathematics

Warm-up: The Basics

Sequence vs. Series

• Sequence: A list of numbers separated by commas. (e.g., \(3, 6, 12, ...\))
• Series: The sum of numbers in a sequence. (e.g., \(3 + 6 + 12 + ...\))

Finite vs. Infinite

• Finite: Has a definite start and end point. No ellipses.
• Infinite: Continues indefinitely, indicated by dots (\(...\)).

Geometric Cheat Sheet

nth Term

\[ a_n = a_1 \cdot r^{n-1} \]

Finds the specific value at position n.

Finite Sum

\[ S_n = \frac{a_1(1-r^n)}{1-r} \]

Sums the first n terms of a sequence.

Infinite Sum

\[ S_{\infty} = \frac{a_1}{1-r} \]

Valid only when \(|r| < 1\).

Practice 1: Classification

Classify: Arith/Geom, Finite/Infinite, Sequence/Series

Timestamps: 23:18 - 26:40

Why Classify First?

"Before you pick up a calculator, you must pick a direction. Using a geometric formula on an arithmetic sequence results in 100% error, no matter how good your algebra is."

Practice 2: Infinite Sums

Calculating the limit of a converging series.

Timestamps: 28:19 - 31:19

The Gatekeeper

Infinite sum formula only exists if:

|r| < 1

"If you apply the formula to a divergent series, you'll get a math 'ghost'—a number that doesn't actually represent the sum."

Your Mission: Error Analysis

You are now the grader. Four students have submitted solutions to complex problems, but each one has made a fatal mathematical error.

Spot the mistake
Explain the logic fail
Provide the correct path

Case File #402

Begin Activity

Lightbulb Moment

"What was one thing you realized today about the importance of checking convergence before calculating?"

Series Sleuths • End of Session

Series Sleuths

Series Sleuths Slides

Error Analysis Worksheet

Error Analysis Answer Key

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