1+2+3+4+... = S

(1+3+5+...)+(2+4+6+...) = S

2S-(1+1+1+1+...)+2S = S

S = 1+2+3+4+...=1+(1+1)+(1+1+1)+...=?1

2S-S+2S=S

3S=S

2S=0

S=0

Try to refute this. This is just as legit, if not more so, than -1/12.