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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It’s hard to ELI5 without losing some mathematical rigor, but what matters is that the amount you add with each “step” of the sum is less than the difference between the current sum and the value that sum converges to.
Let’s take a variant of Zeno’s paradox. To walk to the flag at the end of a race, I must first cover half the distance of the way there. But before I can finish the race I must first cover half of the distance remaining. But then before I can finish the race I must cover half of *that* distance remaining, and so on.
Do this an infinite number of times, and you reach the flagpole. “Eventually” over an infinite number of increasingly smaller divisions the distance between you and the flag becomes zero, and thus you have completed the race. However you don’t *cross* the flagpole,
This is exactly what happens when you add up 1 + 1/2 + 1/4 + 1/8…. Notice each step takes you half of the distance towards two. Of course it doesn’t have to be half, just small enough that you never cross the limit.
As for your other question, there is nothing that says it applies to terms that are less than one. For example 1000 + 500 + 250 + 125 + 62.5… will eventually converge on 2000. This is the same principle as before, just with larger numbers.
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