Some time ago, I posted a fractions puzzle involving flat bread. You can read about it here. Here’s the question – if I take a flat piece of bread, and give you half, then you give me back a quarter, then I give you back an eighth, and you give me back 1/16, and so on ad infinitum, how much of the original flat bread do I have?

If you’ve studied geometric series, this is an easy problem – but what if you haven’t? Is there any way to find the answer?Can we know what 1 – 1/2 + 1/4 – 1/8 + 1/16 – … adds up to?

To quote Bob the Builder and Barack Obama – yes we can!

Here’s how you do it. Suppose I’m there, exchanging bits of bread with you.

• First, I think “I start with 1 piece of bread, then I lose 1/2, then I gain 1/4, then I lose 1/8, and so on. You end up with some fraction X of the bread, and I get 1-X”
• Then, I imagine “What if we swapped roles? Then you end up with 1-X of the bread and I end up with X of the bread. To get my X, I first receive 1/2 of the bread, then give you 1/4, then take back 1/8, then give up 1/16, and so on”
• Then I imagine “What if we started with twice as much bread? So you start with 2 pieces of bread, then you give me 1. Then I give up 1/2, and take 1/4, then give up 1/8, and so forth. Since all amounts of bread are doubled, I now have 2X”
• However, as far as I’m concerned, this is exactly the same situation as I described at the start – I start with 1 piece of bread, then I lose 1/2, then gain 1/4, then lose 1/8 and so forth. We’ve just figured out that this gives me 2X of bread, where X is what it gives you.

So, this procedure gives me twice as much bread as you. Since there’s only 1 piece of bread, I must have 2/3, and you have 1/3.

If you understand this trick, you are partway towards understanding geometric series generally.

Can you use a similar trick to calculate 1 – 1/3 + 1/9 – 1/27 + 1/81 – … ?