He then said that some of you may be surprised to know that $\sum _{n=1}^{\infty }n$ is actually negative - in fact it's $-1/12$. If you substitute it in you get $D=26$ as required.

I think this is amazingly beautiful! Not only is it nice that something like $\sum _{n=1}^{\infty }n$ is evaluated as $-1/12$, but it is rather wonderful that you can derive a dimension (something discrete) by an infinite sum. Usually infinite sums give you continuous quantities (infinite sums are after all about analysis and limits) but here it gives the dimension. There is no reason $D$ even had to come out as an integer, but it does!

(Note to anyone studying analysis: of course we're not claiming that $\sum _{n=1}^{\infty }n=-1/12$ is literally true. Under the standard definitions of analysis it is not true - the left hand side is infinity. However there is a strong intuitive sense in which it is true - and this is what's relevant to string theory. For example if you look at $\sum _{n=1}^{\infty }n{e}^{-n\lambda }$ where $\lambda >0$ then you get:

$A{\lambda }^{-1}-1/12+O(\lambda )$

for some constant $A$. If you then subtract off the infinite part, $A{\lambda }^{-1}$, then take the limit $\lambda \to 0$ you get the answer $-1/12$.

This ''subtracting infinite parts off'' occurs all the time in quantum field theory, basically because the energy of the vacuum is infinite!

An alternative way of getting $\sum _{n=1}^{\infty }n=-1/12$ is to look at the zeta function. The zeta function is defined by:

$\zeta (z)=\sum _{n=1}^{\infty }{n}^{-z}$

for any complex number $z$ with $\Re z>1$ (so that the right hand side converges). Now you can prove that there is a unique way of extending this function so that it is analytic on the whole complex plane (except $z=1$). If you do this you find that $\zeta (-1)=-1/12$. Plugging this back into the original definition, we have the formal result:

$\sum _{n=1}^{\infty }n=-1/12$