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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If you want a physical example with something you might have experience manipulating with your own hands, think about paper sizes. Specifically the A0, A1, A2, A3 etc. ones as you see in the second image here [https://en.wikipedia.org/wiki/Paper_size](https://en.wikipedia.org/wiki/Paper_size).
The idea is that A1 is half the size of A0, A2 is half the size of A1 etc.
So take a sheet of A0 paper.
Cut it in half to get 2 A1 sheets and put one on a table.
Take the remainder (now also of size A1), and cut it in half to get 2 A2 sheets and put one on the table.
Etc. Etc.
Two things should be clear if you keep doing this pattern (or series):
1. We will never run out of paper – we only ever put down half of what we have left. So this series is *infinite*. There is no “end”.
2. The amount of paper on the table must be less than the original amount we started with because we always have some left. And so the sum of those we have placed is *finite*.
We can get arbitrarily close to putting down all of the paper we started with by completing the pattern up to any paper size you want, e.g. A10000. And if that is not close enough we can just go further along the series, e.g. A10001, A100002 etc. By this method we can put down more than any amount of paper which is less than the original amount. But we can never actually put all the paper down.
This, basically, is what a limit is, and the value of that limit is finite – it’s the amount of paper we started with.
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