Come on! Certainly you would think the answer is INFINITY
But here is an amazing proof that the answer = -1/12!!
S = 1+ 2 + 3 + 4 + 5 + 6.............. = -1/12
S1 = 1-1+1-1+1-1+1.... (GRANDI'S SERIES)
if this series ends at an even number ans = 0
but if it ends on an odd number ans = 1
Therefore we take the average = (0+1)/2 = 1/2
S2= 1-2+3-4+5-6+7-8..........
S2 + S2 = 1-2+3-4+5-6+7-8..........
+1-2+3-4+5-6+7-8..........
2S2 = 1-1+1-1+1-1+1......... (which we know = 1/2)
2S2 = 1/2
S2= 1/4
S-S2 = 1+2+3+4+5+6+7+8+9........
-(1-2+3-4+5-6+7-8..........)
S-1/4 = 4+8+12+16+20.........
S-1/4 = 4(1+2+3+4+5+6....)
S-1/4=4(S)
-1/4=3S
S = -1/12
BOOM!
Comments