Is the argument used by the Numberphile video to show that $1 + 2 + 3 + 4 + \dots = -1/12$ valid? At the end of the Wikipedia article on:
$$1+2+3+4 +\dots$$ 
an argument is present that the sum adds up to $-\frac{1}{12}$.  
Here is the numberphile video. 
Here's my attempt to fill in the details from the argument:
(1)  Let $S_1 = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + \dots$
(2)  Let $S_2 = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + \dots$
(3)  $2S_2 = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + \dots$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\, 1 - 2 + 3 - 4 + 5 - 6 + 7 + \dots$
(4)  $\,\,\,\,\,\,\,\,\,\, = 1 -1 + 1 -1 + 1 - 1 + 1 - 1 + \dots$
(5)  $S_1$ is the Grandi's series whose Cesaro's sum is $1/2$
(6)  So, $S_2 = 1/4$
(7)  Let $S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \dots$
(8)  $S - S_2 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \dots$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-1 + 2 - 3 + 4 - 5 + 6 -7 + 8 + \dots$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= 0 + 4 + 0 + 8 + 0 + 12 + 0 + 16 + \dots$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= 4 + 8 + 12 + 16 + 20 + 24+ 28 + \dots$
(9)  $4S = 4 + 8 + 12 + 16 + 20 + 24 + 28 + \dots$
(10)  $S - S_2 = S - 1/4 = 4S$
(11)  Solving for $S - 1/4 = 4S$ gives $S = -\frac{1}{12}$
Is this argument valid?  Is there a flaw in the logic that leads to the right answer in the wrong way?  

We need to be very careful with blind manipulations like the ones in the video - in particular, you can get situations like:
$$S = 1 + 2 + 3 + 4 + \dots$$
$$-S = S - 2S = (1 + 2 + 3 + 4 + \dots) - (2 + 4 + 6 + 8 + \dots)$$
$$= 1 + (2 - 2) + (3 - 4) + (4 - 6) + (5 - 8) + \dots = 1 + 0 - 1 - 2 - 3 - 4 +\dots = 1 - S$$
So $-S = 1- S$ and $0 = 1$.
