Extra problem solving

What is the sum of all natural numbers?
S = 1 + 2 + 3 + 4 + ... = ?


Pick S1 = 1 - 1 + 1 - 1 + ...
Then S1 + S1 = (1 - 1 + 1 - 1 + 1 ...)
                  +(1 - 1 + 1 - 1 + ...) # shift one spot
Then 2S1 = 1 + 0 + 0 + 0 + ...
         = 1
Then S1  = 1/2
Pick S2  = 1 - 2 + 3 - 4 + 5 ...
Then 2S2 = 1 - 2 + 3 - 4 + 5 ...
             + 1 - 2 + 3 - 4 ...    # shift one spot
         = 1 - 1 + 1 - 1 + 1 ...
         = S1 = 1/2
Then S2  = 1/4
Then S - S2 = 1 + 2 + 3 + 4 + 5 + 6 + ...
            -(1 - 2 + 3 - 4 + 5 - 6 + ...)
            = 0 + 4 + 0 + 8 + 0 + 12 + ...
            = 4(1 + 2 + 3 + 4 + ...)
            = 4S
Then 3S = -S2
Then S = -1/4 / 3 = -1/12
Conclude S = -1/12

This is an astonishing result. As common sense, the value of S should go to infinity. However it give us -1/12. I suggest this -1/12 should replace 42 to be the meaning of life. And this proof is actually used in modern physics for example in string theory and calculating Casimir effect.

resource:
https://www.youtube.com/watch?v=w-I6XTVZXww&feature=youtu.be
http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF