It may be said that the sequence:

1/2 + 1/4 + 1/8 ... + 1/2^n tends to one as x tends to infinity.

Your problem can be compared another famous problem.

Lets say that we build a lamp that stays on for 30 seconds, then goes off for 15 seconds, then comes back on for 7.5 seconds, and so on.

After a minute, will the lamp be on or off?