What blew my mind is this. What is the sum of the infinite series

1, -1, 1, -1, ...

One answer is to look at it like this:

(1 - 1) + (1 - 1) + ... = 0

Another answer is to look at it like this:

1 + (-1 + 1) + (-1 + 1) + ... = 1

But then it gets weirder. What if you add two of the series together like so:

1 + -1 + 1 + -1 + ...

____ 1 + -1 + 1 + ...

(Please ignore the underscores. They're just there because otherwise Lemmy messes up the whitespace.)

All the terms cancel out except that first 1 again. But this time it's the sum of two of these series, which means that the sum of one series is 0.5 and somehow not an integer.

The correct answer is that you're not allowed to add up infinite series like this so that's why you get contradictory results if you try.