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But, I am confused by his logic. I get lost at 2S.

He says
S=1-2+3-4+5-6+7+...(infinity).
Then he says we want to multiply S by itself.

When he does this, he seems to use the commutative law, which doesn't work because of the subtraction. So, he says:
2S=
1-2+3-4+5-6+7+...(infinity)
...1-2+3-4+5-6+7+...(infinity)

I used the first set of ellipses to move the problem down as he does.

He then states that this equals
1-1+1-1+1-1+1...(infinity) which equals 1/2. This makes sense. However, his answer of 1-1+1-1+1-1+1...(infinity) does not.

If you were to take 2S, you couldn't slide it down because the commutative law doesn't apply to subtraction. Therefore, because he subtracts, he changes the result.

Watch the video. Does anyone see what I am seeing? Or am I misunderstanding it?

For example, let's suppose, instead of going on for infinity, we stop where he stopped the problem:

1-2+3-4+5-6+7
...1-2+3-4+5-6+7
If we add these the way he added them, we do get 1-1+1-1+1-1+1. In this instance, we would not get 1/2 if we averaged the numbers. We would get 1/7 or 0.1428571428571429

Now, if we add it without using the commutative property we get

1-2+3-4+5-6+7 1-2+3-4+5-6+7
2-4+6-8+10-12+14= 8

If we average the answer we get 8/7 or 1.142857142857143

So, let's look at this.

Our answers:
0.1428571428571429
1.142857142857143

If we assume the calculator rounded the last decimal place to 3 on the second answer, he is still a tenth of a digit off. I'd much rather have $1.14 vs $0.14.

Does anyone follow the video? I also have no clue why he chose to take that answer and apply it to his third sum to get -1/12.

I'm no mathematician, but I see very poor logic being applied. I truly hope I misunderstand this, or that they really don't use this in physics.

"For small creatures such as we the vastness is bearable only through love."
- Carl Sagan