1 / infinity

To expand just slightly on Mark's post, the literal answer to "what is 1/infinity" is "an undefined quantity" (as is n/0).

One of the problems with allowing these undefined quantities to be used as though they were numbers is that by using them you can readily prove that any number is the same as any other number.

 

If you can't use infinity in arithmetic expressions, can you substitute things equal to infinity that behave well in those expressions?
Like this? :)

1+2+3... the partial sums of the infinite series grow without bound = infinity
1+2+3...can be summed to equal -1/12
if two things equal to the same thing are themselves equal
then
1/infinity = 1/(-1/12) = -12