If I start at 1 and my goal is to get to 2, I could move to 1.1 and be a little bit closer to 2. Then move to 1.11, and be even closer. Then 1.111, so on and so forth. I understand that with each move, the step forward is smaller than the previous one. But it’s moving forward nonetheless. How is it that I can forever move forward, but never get there?
In: Mathematics
If you reframe your question to begin: “If I start at 1, and my goal is to get to 10/9”, then you’ll perfectly describe [Zeno’s paradox](https://en.wikipedia.org/wiki/Zeno%27s_paradoxes). (10/9 is 1.11111111…)
The more usual form of that paradox discusses starting at 0, goal is 1, and moving forward by 50% of the remaining distance every time, eg terms ending on 1/2, 3/4, 7/8, 15/16, 31/32, and so on. This describes an infinite series that is said to “converge to 1”, and yours converges instead to 10/9.
There are a number of infinite series that converge to a number: despite always increasing, they’ll never increase enough. There’s also series that converge to a number despite alternating increasing and decreasing, such as the [alternating harmonic series](https://en.wikipedia.org/wiki/Harmonic_sum#Alternating_harmonic_series).
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