Fun with infinity

I’m lucky enough to teach two top set year 8 classes (year 7 last year) who are at least semi-geeky about their maths. Last summer I did a lesson with them on the nature and craziness of infinity, with the able assistance of the amazing Vi Hart (or at least her Youtube vids), and Jordan Ellenberg’s book: How Not to Be Wrong. They loved it, so I’ve reproduced some of the resources below.

Vi Hart videos:

The infinite series paradox (1):

What is the sum of: 1 + 2 + 4 + 8 + 16 + …?

To answer this, first multiply the sum by 2: 2(1 + 2 + 4 + 8 + 16 + …) = 2 + 4 + 8 + 16 + …

Now: 2(1 + 2 + 4 + 8 + 16 + …) – 1(1 + 2 + 4 + 8 + 16 + …) = 2 + 4 + 8 + 16 + … – 1 – 2 – 4 – 8 – 16 – … = -1

But the LHS simplifies to: 1 + 2 + 4 + 8 + 16 + … = -1

Thus we seem to have proved that the infinite sum of ever-increasing terms is -1. Huh.

The infinite series paradox (2):

What is the value of the infinite sum 1 – 1 + 1 – 1 + 1 – 1 + …?

Answer 1: (1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0 + … = 0

So far so good. But hang on…

Answer 2: 1 – (1 – 1) – (1 – 1) – (1 – 1) … = 1 – 0 – 0 – 0 … = 1. Huh.

Answer 3: Suppose T is the value of our mystery sum: T = 1 – 1 + 1 – 1 + 1 – 1 + …

Take the negative of both sides: – T = – 1 + 1 – 1 + 1 …

But this is just what you’d get if you take 1 away from T i.e. T – 1.

So T – 1 = -T, which is only satisfied when T = 1/2. Eek.

I find answer 3 both most counterintuitive and most satisfying. Brilliantly, some of my year 7s came up with this solution themselves.

 

Relevance to the classroom:

This stuff isn’t going to be on any curriculum, and one observer in my classroom was heard afterwards to mutter ‘he just let them doodle!’ But for some of our most gifted mathematicians, we have to squeeze in real maths around the staid content of the GCSE curriculum – it might just inspire them to carry on with the subject to A-level and beyond. Plus everyone loves to have their brains messed with.

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