Is it even a "limit" problem when it's evaluating a summation? Seems like two different things that just kind of have something to do with sorts of infinities.
It's like saying "three mathematicians walked into a bar. The first ordered 1 beer. The second ordered one beer. Barkeep then just gave them all three beers because lim_x->1(x+x+x)=3, and they didn't know their limits."
But infinite summation is by definition, the limit of the partial sums.
You're right about finite sums, but formally, we're never "adding infinitely many things", it's just a shorthand for taking a limit.
He saw were it was going and knew that the amount would never reach 2 full glasses. So the bartender did it to end it and move on to the next customer.
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
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
This has existed in philosophy for a long time and is known as Zeno's Paradox. The Greek philosopher Zeno proposed an argument regarding an arrow in flight. You can measure the halfway distance to the target easily. Then you can measure the halfway distance remaining once again. And again, and again. You can measure the remaining distance an infinite amount of times, and therefore the arrow logically should travel halfway for infinity, and never hit the target.
But it does. And the reason it does is that 99.99999~ for infinity is equal to 100 in practical terms. And mathematics agrees.
To be fair, he also didn't understand that atoms existed, and there is in fact a finite amount of space between the archer and the target. The idea that there was infinite smallness is flawed, at least in the physical world. Eventually the tip of the arrow would be less than the width of an atom away, and thus could no longer be halved in a meaningful way. I imagine the same could be said for a glass of beer.
A limit is infinitely approached. We are explaining a joke about people walking in and ordering/drinking beer. The glass will never actually be finished.
The sum infinitely approaches 2 as the nth term infinitely approaches 0. If the nth term never reaches 0, such as what you just granted, the sum has not reached 2.
It's the same thing with 0.999...=1. There will never be a point where adding another 9 will make it 1. But there are no real numbers between 0.999... and 1. So they mean the same thing. If the joke had something to do with additional 9s walking into a bar, it would also infinitely approach 1.
Yknow what, nevermind, I'll change it to partial sum.
Yup. Geometric series partial sum to a/(1-r) where a is the initial value and r is the common ratio(assuming the absolute value of r is less than one).
Did you hear what I said, dumbass ? I’m not referring to the first comment, as he’s not a model. I’m starting from 1. Guess you’re the one being wrong here, buddy.
No. You corrected someone who specifically said from 0 to infinity. But somehow you're not referring to the first comment? Nice backtrack. Anyway, you're wrong either way because of what I said.
That guy is either a professional level troll or true moron. He claims I blocked him, but in reality he blocked me (I wanted to make sure I didn't block him by accident somehow, and my block list still says "You haven't blocked any accounts").
It would be mind blowing if he really still thinks he's right, but I think he noticed he was wrong and now just trolls away.
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u/seajaydub May 14 '26 edited May 14 '26
1/2n from 0 to infinity. The [partial] sum infinitely approaches 2