If S=1+2+3+4+5+6.... infinity and T=1-2+3-4+5-6... infinity and C=1-1+1-1+1-1... infinity we know...

C=1,0,1,0,1,0,1,0,1... infinity, average the two values, C=1/2

2T=1-2+3-4+5-6... infinity
+1-2+3-4+5... infinity
therefore, 2T=1-1+1-1+1... infinity, so 2T=C=1/2
therefore T=1/4

S-T=1+2+3+4+5... infinity
-(1-2+3-4+5... infinity)
S-T=4+8+12+16... infinity
S-T=4(1+2+3+4+5... infinity)
S-1/4=4S
-1/4=3S
-1/12=S

THEREFORE the sum of all natural numbers S is equivalent to the value -1/12

QED