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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Some infinite sums like the example you gave are relatively easy to wrap your head around in different ways.
Depending on how your brain is wired, one of the easiest for that is to simply graph it.
You can tell that you get a curve that will approach 2 the longer you add things and end up at 2 in infinity.
You can also visualize it by stacking rectangles of the appropriate size. You have one square that is 1 by 1, one rectangle that is 1 by 0.5 one that is 0.5 by 0.5 and so on and as you draw them you will find that you end up with a rectangle that is 1 by 2 with a series of increasingly tiny boxes in whatever corner you stacked them.
You can also look at the fractions that come out at every step: 1/1, 3/2, 7/4, 15/8, 31/16, 63/32 … and see that this gets ever and ever closer to just being two.
Other infinite sums are just weird and counterintuitive and sum up in really unexpected ways that are hard to visualize. Some don’t have any solution at all.
It can get pretty out there.
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