There is a Wikipedia page for that serie which uses analytic continuation to obtain -1/2.

S1=1+1+1+1+...
S2= 1-1+1-1+...
S1+S2= 1+2+0+2+0+...
Only sum is shifted like what is done for other series summation.

And finally for more fun:
S1=-1/2
S1/6=-1/12
So:
1/6+1/6+1/6+1/6+... = 1+2+3+4+...

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