mathematics

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

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

   = 1 - 1 + 5 - 1 + 9 - 1 + 13 - 1 + ...
   = (1 - 1) + (5 - 1) + (9 - 1) + (13 - 1) + ...
   = 0 + 4 + 8 + 12 + ...
   = 4 * (1 + 2 + 3 + ...)
   = 4 * S

so,
S+T = 4S
T = 3S

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

Then,
S-T = 0 + 4 + 0 + 8 + 0 + 12 + ...
    = 4 * (1 + 2 + 3 + ...)
    = 4 * S

so,
S-T = 4S
-T = 3S


Combining two equations
T = 3S
-T = 3S

T = -T ==> T=S=0

so,
1 + 2 + 3 + 4 + 5 + 6 + ... = 0
1 - 2 + 3 - 4 + 5 - 6 + ... = 0