does this apply only for terms that are less than 1 eg 1 + 1/2 + 1/4…. or does this apply to all ap/gp. I remember studying this, but it’s been so long I remember only the gist.
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Think of a situation where you are in a room. You can take one step at a time, but each step is half of the distance between where you are and the other side of the room. It is important to note: another way to say this is your first step is half of the room, and every step after is half the distance of the previous step but we will come back to that. Now, mathematically speaking, since you are always getting halfway between where you are and the end, you will never reach the end, but you will keep getting closer and closer to a single spot no matter how many times you step. So an infinite number of distances still converge to a single distance value.
Now this only works if the absolute value of the change in terms generally gets smaller over time. (this only works if you are moving towards a spot, sometimes you can go past the spot, if you are guaranteed to go back and forth over the spot getting closer and closer each time) for example, imagine you are on a football field standing at one end. Every step you take is 1.5 times the distance between you and the 50 yard line in the direction of the 50 yard line. You would go from 0, to 75, to 37.5, etc. You would keep going back and forth over the 50 but each step you would be stopping closer and closer to the 50.
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