How does the sum of 1/2 + 1/4 + 1/8 etc. (Halves each time) eventually become 1? I understand it’s infinite but surely it still wouldn’t work?
In: Mathematics
The key here is the Archimedean property of real numbers which states the following: for any positive number x, however small, there exists a large enough natural number n such that x>1/n.
Imagine you subtract 1/2, then 1/4, then 1/8 and so on from 1. If we assume there is a number x we are left with after this process finishes, the number x is less than 1/2, less than 1/4, less than 1/8 and so on.
That means I can show x is simultaneously less than any number 1/n and therefore it’s not *positive*. Hence it is zero. (I don’t think I need to convince you it’s not negative).
For the general idea, imagine you have a plank one meter long, and you take half of it, add 1/4 which is half of the half left, then again half of what was left again, and so on, you can “sense” that it all adds up to the complete plank if you go on long enough (you’re adding slices of atoms).
For the math, think about it the other way. How do you prove 0.99999999… is different from 1? Their difference would not be zero, let’s say “x”. Whatever x might be, you can show the real difference is even smaller, again and again. The only value that works is if x=0, so they are the same.
Note: it’s hard to stick to the “like I’m 5” rule!
You’re describing an infinite series here. This series defines a sequence with the values 1/2, 3/4, 7/8, etc.
Now, you’re interested in how this series behaves if you follow it all the way to infinity. This is called a limit. This means that the sum of that series being 1 means that it becomes indistinguishable from 1. Generally speaking, mathematicians have agreed that a series only eventually becomes a certain number if it becomes indistinguishable from it.
So what do I mean by indistinguishable? We generally consider two numbers to be the same if there’s no other number between the two. One common example is 0.999… = 1, because any number between the two would have to be either 0.999… or 1 itself.
We can extend this notion to limits. So, it’s obvious that 1/2 + 1/4 + 1/8 etc. never becomes bigger than 1. So, it’s left to show that it’s not smaller than 1 either. But if you assume that the limit is smaller than 1, you’ll find that the series always eventually becomes bigger than that supposed limit at some point, no matter which number smaller than 1 you chose. So you have that the limit can’t be bigger than 1 and it also can’t be any number smaller than 1. The only remaining option is that it’s equal to 1.
[deleted]
Idk how to explain calculus like you’re five but I’ll try.
Basically, it doesn’t actually become 1. Instead the addition of 1/2 + 1/4 + 1/8 + . . . . . 1/2n becomes infinitely close to 1, to the point where it’s pretty much just 1.
Imagine eating a cupcake, but leaving behind a crumb. You’ve still eaten a cupcake. Similarly, this sum is still considered 1 even though it’s not quite 1.
In calculus or precalculus this would be a limits problem. Given that you’re trying to find the sum of 1/2 + 1/4 + 1/8 + . . . 1/2n, as n approaches infinity, the sum approaches 1.
I hope this helped? If it didn’t I’m sorry and someone better at math than me will probably come by and explain better
Congrats, you just stumbled across Zeno’s Paradox.(Kinda)
The short answer is because infinity. When you are adding half of something infinitely you approach an infinitely small portion so it ceases to be useful to measure.
Other comments are correct, regarding the calculus proofs. But you can kind of think of it like this.
While the numbers you add up are getting infinitely smaller, you are adding them up infinite times.
It kinda is easier to imagine things as infinitely small rather than things as being done an infinite amount of time. However, the maths equation doesn’t care about that, the two infinite processes are working against each other and kind of cancel out to a standard number that we are comfortable with.
In this case, it’s easy to evaluate the infinite series.
S = 1/2 + 1/4 + 1/8 + …
Multiply both sides of the equation by 2:
2S = 2/2 + 2/4 + 2/8 + …
2S = 1 + 1/2 + 1/4 + 1/8 + …
So
2S = 1 + S
Subtract S from both sides, and
S = 1.
Latest Answers