If 1/infinity is the number that when multiplied by infinity equals 1, then the number is 1/infinity.
To arrive at you would multiply (infinity)*(1/infinity)=infinity/infinity which does technically equal 1, but it does also technically equal 2 or and any other positive real. There 1/infinity should be treated as undefined.
You also need to distinguish between the use of the entity
[math]\frac{1}{\infty }[/math]
in loose form by applied mathematicians (like me)
and proper limits like this
[math]\mathop {\lim }\limits_{x \to \infty } \frac{1}{{{x^n}}};x > 1,n \ge 0[/math]
[math]\mathop {\lim }\limits_{n \to \infty } \frac{1}{{{x^n}}};x > 1,n \ge 0[/math]
where if you pick the debarred values of x or n you will converge to 1 or 1/0