Level 4: Endless sums and fractals

Infinity and strange math · level 4 of 8 · rough age 11-12 (a guide for parents; anyone can start anywhere)

In level 3 you met Zeno's race, where infinitely many stages fitted into a few seconds. In this level you learn to add up infinitely many numbers: some endless sums settle on an exact total (like \(\frac12 + \frac14 + \frac18 + \cdots = 1\), or \(0.999\ldots = 1\)), while others grow past every number, even though their pieces shrink. Then you build shapes by repeating a step forever, the fractals: a triangle with no area left, and a snowflake with an endless edge around a finite area. The level ends back at Hilbert's hotel, with infinitely many buses.

Lesson	You will be able to
Halves that add up to 1	Prove with a picture and with partial sums that \(\frac12 + \frac14 + \frac18 + \cdots = 1\), and add other endless sums the same way
Does 0.999... equal 1?	Give three different arguments that \(0.999\ldots = 1\), and explain why it feels wrong
Sums that blow up	Prove that \(1 + \frac12 + \frac13 + \frac14 + \cdots\) grows without limit, and use it to stack books out past a table edge
The Sierpinski triangle	Count the triangles at each step, and show that the area left shrinks to zero
The Koch snowflake	Show that the snowflake's edge grows without limit while its area settles at \(\frac85\) of the first triangle
Hilbert's hotel with infinitely many buses	Give every passenger of infinitely many buses a room using powers of primes, and see why that makes a set countable

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