Question

Why and how is 1,2,3,4,5....∞ = -1/12?

Answer 1

let,
S1= 1-1+1-1+1-1+1-1..............
then If last term is +1 then the value of S1 will be 1 and if last term is -1 then value of S1 will be 0
hence, we will take S1 as their average that is 1/2
now take another series ,
S2=1-2+3-4+5-6.......
+
S2= 1-2+3-4+5-6.....
=
2S2=1+(-2+1)+(3-2).....
2S2=1-1+1-1+1-1....
2S2=S1
2S2=1/2
S2=1/4
Let us take another series S3=1+2+3+4+5.......
-
S2=1-2+3-4+5-6
S3-S2=4+8+12...
S3-S2=4(1+2+3....)
S3-S2=4S3
-S2=3S3
-1/4=3S3
S3=-1/12


hence proved, it is used in all physics concepts including string theory

Answer 2

The first one, I wrote it years back.