Taming Infinity

Warning: Math Content!

So the other day I saw this very cool mathematics trick. Suppose you have a divergent series (a group of numbers that adds up to infinity) that adds up like this:

1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + ...

Where the next number in the series equals to twice the value of the previous number.

It is pretty obvious that when you have infinite numbers in this series, this adds up to infinity right? Well, yes. But, it can also be mathematically proven that the whole series equals to -1.

What?

I know. Pretty mind-boggling stuff. But watch the video below, it explains how the series can become -1 by some mathematical manipulation



So, to reiterate what the video just said. Suppose you have the series

1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + ... .

Now the whole series will be the same if you multiply it with 1, right? Anything that is multiplied by 1 equals to the original value. Like this

1 * (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + ... )

It wouldn't change a thing. Now we all know 1 = 2 - 1, so we can substitude 1 in front of the series with (2 - 1), which then becomes

(2 - 1) * (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + ... ).

Now we expand the brackets by multiplying 2 and -1 in (2 - 1) separately with the series, which becomes

( 2 + 4 + 8 + 16 + 32 + 64 + 128 + ... - 1 - 2 - 4 - 8 - 16 - 32 - 64 - 128 - ... ).

If you noticed, except for - 1, the other numbers on the right side can subtract the numbers with + sign. And in the end, you will be left with only -1!

So... 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + ... = -1!

Isn't it amazing? Although I do not know how does this apply in anything because I am not a mathematician, I still think it is very interesting how some problems in mathematics can have answers that are so counter-intuitive, and not to mention aesthetically simple too (at least that is what I think).



Oh well. Till next time, cheers~