What are the basic rules for manipulating diverging infinite series? This is something that I played around with in Calc II, and it really confuses me:
$s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots = \infty$
$s - s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots $
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -  (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots) $
$s - s = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + \ldots$
$0s \ \ \ \ \ = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + \ldots = \infty$
$\therefore 0 (\infty) = \infty$
I'm aware the $0\cdot\infty$ is indeterminate, so what am I doing wrong? What are the basic rules for manipulating these diverging infinite series? 

Basically, you don't. 
The only way to manipulate them is if you know precisely what you're doing and using renormalization techniques or analytic continuation or something, but those techniques won't be taught for awhile and have specific rules about them.
Long story short, you don't manipulate them.
