TITLE: $S=1+10+100+100+10000+... = -1/9$? How is that
QUESTION [15 upvotes]: its written that if you multiply the sum above by -9 and use the distributive law, all terms except "1" will cancel. I can't see that. I know that this is a divergent series. (the article I was reading was a layman's introduction to zeta regularization)  
Similarly, how is $S=1+2+3+4... = -1/12$
Even if the series were to terminate somewhere, I dont see these value. This is completely baffling me.
Using $\frac{1}{1-x}$ I can put x=10 but that is not fair,isn't it?

REPLY [3 votes]: for your $1+2+3+4+...=-1/12$ this is the value of $\zeta(-1)$, the riemann zeta function at $-1$. so while it is not an equality as written, the definition is usually given as $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ which only converges for real $s$ greater than 1.  however it can be continued to the rest of the plane (with a pole at 1).