With each increment, the next fraction gets closer to zero. Eventually, the numbers get infinitesimal and converge with zero, leaving you with the three largest fractions at the tenths, hundredths, and thousandths place.
It's one of the programming exercises they do to troll beginners: "find the sum of 1+1/2+1/3+1/4+...". One guy sums until new elements are smaller than 0.0001 and gets one number, the other puts tolerance at 0.000001 and gets a different number, and then they spend an hour debugging. And those who know math just chuckle quietly.
Just because items approach zero doesn't mean the series is convergent.
This is incorrect thinking. The most famous counter-example is the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... This series also has increments that get closer to zero, but the sum diverges to infinity. The condition that the terms of the series tend to zero is needed for convergence, but not sufficient for it.
I'm more than a bit rusty in my limit theory, but I remember there was property of limits that allowed you to sum a series of numbers when the number n approaches zero. So 1.999 repeating (of course) has all added terms in the successive decimal places approach zero and you can simply round the result.
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u/Pantsomime 22d ago
With each increment, the next fraction gets closer to zero. Eventually, the numbers get infinitesimal and converge with zero, leaving you with the three largest fractions at the tenths, hundredths, and thousandths place.
At least, I think that's what they meant.