That's what it says in the original post. It is also non-sensical. If the probability is 10% (p=0.1), and the number of systems is 11 (n=11), then you get 1/(1 - 11*0.1) which is -10.

I tried working this out, but I think the original was correct.

The cases where the answer is negative correspond to a 'runaway scenario' where every change is expected to cause more than 1 extra change. So the answer is 'nonsensical' (because that is indeed where the formula for geometric series no longer works) but the true answer is infinity.

Correct!

Valid for n*p < 1, geometric series convergence, I should have mentioned.

It is easy to see this by forming the geometric series and rearranging terms, the standard Euler trick.