mathematics

Let,
R =  1 + 2 + 3 + 4 + 5 + 6 + ...
S =  1 - 1 + 1 - 1 + 1 - 1 + ...
T =  1 - 2 + 3 - 4 + 5 - 6 + ...

Consider 1-S,
1-S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + ...)
    = 1 - 1 + 1 - 1 + 1 - 1 + ...
    = S

So S = 1/2

Also,
2T = 1 - 2 + 3 - 4 + 5 - 6 + ...
   +     1 - 2 + 3 - 4 + 5 + ...
   = 1 - 1 + 1 - 1 + 1 - 1 + ...
   = S = 1/2

So T = S/2 = 1/4

Next,
R - T = 1 + 2 + 3 + 4 + 5 + 6 + ...
      - 1 + 2 - 3 + 4 - 5 + 6 + ...
	0 + 4 + 0 + 8 + 0 + 12 + ...
      = 4 * ( 1 + 2 + 3 + 4 + ...)
      = 4 * R
So,
3 * R = -T
R = -T/3 = -(1/4)/3 = -1/12

Therefore,
1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12