Apparently ramanujan proved it but couldn't believe the proof himself. He wrote to Hardy telling him that if he shows him the proof, Hardy would show him to the lunatic asylum.
The simplified proof is as follows ( at least how I understand it)
Let's say we have as sum S1 defined so
S1 = 1 - 1 + 1 - 1 + 1.......
You could say
S1 = 1 - ( 1 - 1 + 1 - 1....)
S1= 1 - S1
S1=1/2, which is kind of screwy to beging with.. But it's mathematically sound.. This is what happens when you start screwing around with infinite divergent series
Now let's take a different series
So let's say we add S2 to itself but we screw around with how the addition a bit
There are two problems though. 1/2 is only the Cesàro sum of the series. The real sum does not exist, exactly because the series is divergent. The second problem is that you may not simply rearrange terms when you add divergent series.
You only arrive at 2*S2 = 1-1+1-1+1 ... because you're lining up both series, and slightly offsetting one of them. This is valid for convergent series, but not for divergent. Using this method, I can also show that 1-1+1-1+1 ... sums to 6.
The mind is a strange and wonderful thing. I'm not sure that it will ever be able to figure itself out, everything else, maybe. From the atom to the universe, everything, except itself.