Honestly, there is no real answer to 1+2+3+... other than it diverges. Here, watch this: 1+2+3+... = S
1 + (2+3+4) + (5+6+7) + (8+9+10) + ... = S 1 + 3*3 + 3*6 + 3*9 + ... = S
1 + 9*1 + 9*2 + 9*3 + ... = S 1 + 9S = S S = -1/8
And yet...
1+2+3+... = S
1+3+5+7+... + 2+4+6+8+... = S
1+3+5+7+... + 2S = S
1+3+5+7+... = -S
(1+3) + (5+7) + (9+11) + ... = -S
4 + 12 + 20 + ... = -S
4(1 + 3 + 5 + 7 + ...) = -S
-4S = -S
S = 0
oops i wrote the expression as 1/2*n^2 + n instead of 1/2*n^2 + 1/2*n
shame on me
Honestly, there is no real answer to 1+2+3+... other than it diverges. Here, watch this: 1+2+3+... = S
1 + (2+3+4) + (5+6+7) + (8+9+10) + ... = S 1 + 3*3 + 3*6 + 3*9 + ... = S
1 + 9*1 + 9*2 + 9*3 + ... = S 1 + 9S = S S = -1/8
And yet...
1+2+3+... = S
1+3+5+7+... + 2+4+6+8+... = S
1+3+5+7+... + 2S = S
1+3+5+7+... = -S
(1+3) + (5+7) + (9+11) + ... = -S
4 + 12 + 20 + ... = -S
4(1 + 3 + 5 + 7 + ...) = -S
-4S = -S
S = 0
oops i wrote the expression as 1/2*n^2 + n instead of 1/2*n^2 + 1/2*n
shame on me