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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Since you mentioned studying this before, for geometric series (including the one you mentioned) it converges (or adds up to a finite number) if r is between -1 and 1. This is not true “only for terms that are less than 1”; the classic counterexample being the harmonic series (1 + 1/2 + 1/3 + 1/4 + 1/5 …..). There are also infinite series that don’t go to infinity, but also don’t converge, like 1 – 1 + 1 – 1 …..
If your confusion is how adding infinitely many numbers doesn’t go to infinity, ala Zeno’s paradox, it basically comes down to the extra terms are so small they don’t grow the sum to infinity. Let’s consider the infinite series .3 + .03 + .003 + .0003 ….. It’s pretty clear this number will not rise above even .4. In fact, if you remember long division, this sum is just 1/3.
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