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
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.
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