Understanding the summation of diverging series

[whit3r0se-]

I was recently researching into some string theory when i came across the following summation:
The sum of all natural numbers is -1/12, now I'm still wrapping my head around the context of the application within critical string dimensions, but is this summation valid? And if not, why it being used in this way?

 

[micromass]

No, it's not valid. The sum of all natural numbers is infinity, and not -1/12. There are certain ways to assign a number to divergent series however, those ways are called summation methods. You have Abel summation, Cesaro summation, Ramanujan summation, etc. These numbers are not the sums of the series in question. So while $1+2+3+4+...$ can be Ramanujan summed to -1/12, the sum of all natural numbers is still infinity, as it should be.

 

[berkeman]

Thread closed.