A Sum with two answers

What is 1 - 1 + 1 - 1 + 1 - 1 + .... ?

There are two answers to this.

It all depends on how we pair the numbers.

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

Which is correct? The present author takes the view that there are two sensible answers.

The first is that the series oscillates between the values 0 and 1.
The second is that the sum to infinity of the series is a meaningless concept.
Actually, I feel the latter is correct.

What I was not prepared for is that the sum of all positive integers
1 + 2 + 3 + 4 + 5 + 6 + ... is actually -1/12.
Until I read this document, produced by a very clever Year 9 boy:

Sum of the Positive Integers