1 + 2 + 3 + 4 + 5 + 6 + ... = ?
gotta be infinity, right?
let
S = 1 + 2 + 3 + 4 + 5 + 6 + ...
S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...
S2 = 1 - 2 + 3 - 4 + 5 - 6 + ...
add S1 to S1:
S1 *2 = 1 - 1 + 1 - 1 + 1 - 1 + ...
+ 1 - 1 + 1 - 1 + 1 - 1 + ...
= 1
so, S1 = 1/2
add S2 to S2:
S2 *2 = 1 - 2 + 3 - 4 + 5 - 6 + ...
+ 1 - 2 + 3 - 4 + 5 - 6 + ...
= 1 - 1 + 1 - 1 + 1 - 1 + ...
= S1
= 1/2
so, S2 = 1/4
subtract S2 from S
S - S2 = 1 + 2 + 3 + 4 + 5 + 6 + ...
-[1 - 2 + 3 - 4 + 5 - 6 + ...]
= 0 + 4 + 0 + 8 + 0 + 12 + ...
= 4 *(1 + 2 + 3 + 4 + 5 + 6 + ...)
= 4 * S
S - 1/4 = 4 * S
-1/4 = 3S
so, S = -1/12
this is nuts...
Here are the professors explaining it:
