Infinite Horizon

Mastering Infinite Geometric Series

\( \sum_{n=1}^{\infty} a_1 r^{n-1} \)

Zeno's Paradox

A Journey to the Door

Imagine you walk half the distance to a door.

Then, you walk half the remaining distance.

"If you keep doing this, do you ever actually reach the door?"

The Pattern

\( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots \)

Getting closer... forever.

Concept Deep Dive

Convergence & The Sum Formula

Watch For:

The condition for convergence: \( |r| < 1 \)

The Formula:

\( S_{\infty} = \frac{a_1}{1 - r} \)

Does it have an end?

Convergence

The sum approaches a finite value as the number of terms increases.

CONDITION:

\( |r| < 1 \)

Example:

\( 100, 50, 25, 12.5, \dots \)

\( r = 0.5 \) (Converges!)

Divergence

The sum grows without bound (approaches infinity).

CONDITION:

\( |r| \geq 1 \)

Example:

\( 2, 4, 8, 16, \dots \)

\( r = 2 \) (Diverges!)

Guided Practice

Live Solve

Problem 1

\( 100 + 50 + 25 + \dots \)

Identify \( a_1 \): _______

Identify \( r \): _______

Calculate \( S_{\infty} \): _______

Problem 2

\( 2 + 4 + 8 + 16 + \dots \)

Identify \( a_1 \): _______

Identify \( r \): _______

Sum exists? Yes / No

"Infinite terms don't always mean infinite sums. The common ratio \( r \) holds the power!"

Decimal Metamorphosis

Level: Extension

Did you know a repeating decimal is actually an infinite geometric series?

\( 0.333\dots \)

Can we use our sum formula to prove this is exactly \( \frac{1}{3} \)?

The Breakdown

1 \( \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \dots \)
2 \( a_1 = \frac{3}{10} \)
3 \( r = \frac{1}{10} \)
? Calculate \( S_{\infty} \)

Infinite Horizon Mastery

Infinite Horizon Slides

Infinite Series Practice Worksheet

Decimal Metamorphosis Challenge

Series Mastery Cheat Sheet

Infinite Series Teacher Guide

Infinite Series Answer Key

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