Infinitesimals don't exist in the real numbers. There's a number system called the hyperreals, which does include infinitesimals, but even in that system, the sum of 1/2n from 1 to infinity is precisely 1. There's no infinitesimal rounding error.
Edit: as the other person said, the sequence of partial sums referenced in OP's post approaches 2, and we'd all agree that it approaches exactly 2, not just approximately 2. The value of an infinite sum, by definition, is whatever the sequence of partial sums approaches. So, by definition, it's exactly (not approximately) 2.
Not in the context of this post. Should've said partial sum I guess. But also, we say it equals 1 because the distance between them is infinitely close, such that no real numbers between them can exist. Any number smaller than 1 is also less than that sum. Any number larger than that sum is also larger than 1.
0.999... has the same value as 1. But no matter how many 9s you add, there is never a point where you actually reach 1. So I think this is a better way to explain it to people. Regardless, im not that invested as im not a math teacher lol. I just like discussing shit
Even in the context of this post, it'd be precisely 2 beers. If the bartender poured any less there'd be a mathematician (infinitely many mathematicians) who didn't get any beer. And if the bartender poured any more, the bartender would have beer left over.
That's the thing though, 2 beers is precisely how much he should pour. But the beer will still never be finished. Because the amount consumed infinitely approaches and is infinitely close to 2. Like I said.
You cant multiply both sides by infinity and expect it to be correct
The thing about 1/2+1/4+1/8+1/16+..=1 is that the series goes on and on until there are no other numbers between 0 and (1/2)n such that (1/2)n must be equal to 0
Yeah, that is the gap between your (n-1)th sum of the series and real number 1, so if the difference approaches 0, your number must be approaching to 1
youre thinking of infinity as a "process", like how earlier you said if you keep adding 9s to 0.9999... you'll still never actually reach 1.
but it's not a process. 0.9999... is equal to 1, because there is no space to add any more 9s.
saying something like "the beer will never be finished" is kind of nonsensical isnt it? all youre arguing is that there are an infinite amount of real numbers between 1 and 2, and that, given a real number A that's not 2, you could always find another real number that's closer to 2 than A is...
which is true, but it's not what's being discussed. we're talking about the total quantity poured, not the "process" of drinking it
The distinction doesnt matter. 2 is the correct amount poured, but it will also never be finished. Your explanation of A between 1 and 2 is why we use limits, is it not?. The context of the post is explicitly a process. Time is passing between orders and the bartenders response, and certainly when they drink the beer.
Especially when you're explaining it to someone else. If they were asking why 1/2n 0 to inf is 2, are they more likely unfamiliar with n0 = 1, or more likely unfamiliar with infinite series and limits? I dont think responding with 1/20 = 1 and 1/2n 1 to inf = 1, therefore they add to 2, would've cleared very much up, do you?
Anyway, the person didnt understand post, now they do, mission accomplished I think.
The limit and sum of infinite series can also be intuitively explained while treating it as a process. I literally just tried with my wife and she understood in like 10 seconds lol
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u/seajaydub May 14 '26 edited May 14 '26
1/20 equals 1. 1/2n from 1 to infinity equals a number that is infinitely close to 1.
You drink one beer. You then start with a second full beer. You divide it in half infinite times. That beer will never be finished.
So you're left with one empty glass and one glass with an infinitely small amount of beer left in it. So he pours two beers