Question

why does a geometric series equation reach a stabilisation point?

Answer 1

Because at some point you're adding new numbers that are so small that it does not really raise the total sum past a certain number. For example, let's say a series converges to 2. After we get through so many sn's, we get to the point where we might be adding like this:
1.9 +.09 + .009+.0009+.0009+.000009+.0000009. We continue on with our summation to infinity, but all we end up doing is adding decimal places, which is why we can say an infinite series actually does reach a stabilization point.

Answer 2

it only reaches a sublimation point if r <1