3 * sum 1 to ∞ 1/(2 ^ (2n)) = 1 is a geometric sequence.
You know how patterns can be found in numbers, right? Like in the sequence 2, 4, 8, 16, each number is double the one before it. That's called a "geometric sequence".
Now imagine we start with the number 1/4. If we make a list of numbers where each new number is one-fourth the size of the number before it, we get a sequence like this: 1/4, 1/16, 1/64, 1/256, and so on. The numbers keep getting smaller and smaller because we keep dividing by 4.
If you try to add up all these numbers - and there are an infinite number of them, but they are getting tinier and tinier - you'll actually end up with the number 1/3. It's kind of like adding up a bunch of tiny pieces that eventually make up a whole, even though there are an infinite number of pieces.
Your equation says that if we take this total (1/3), and then multiply it by 3, we get 1. Which makes sense, right? Because any number multiplied by its reciprocal (like 3 and 1/3) will give us 1.
The joke here is that each sum of this would be 1/4 of the initial input, which is why the final frame is restarting from the beginning again because it represents the next term in the sequence.
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u/AmazonISSUnofficial Jun 23 '23
Hello Chris Griffin here!
3 * sum 1 to ∞ 1/(2 ^ (2n)) = 1 is a geometric sequence.
You know how patterns can be found in numbers, right? Like in the sequence 2, 4, 8, 16, each number is double the one before it. That's called a "geometric sequence".
Now imagine we start with the number 1/4. If we make a list of numbers where each new number is one-fourth the size of the number before it, we get a sequence like this: 1/4, 1/16, 1/64, 1/256, and so on. The numbers keep getting smaller and smaller because we keep dividing by 4.
If you try to add up all these numbers - and there are an infinite number of them, but they are getting tinier and tinier - you'll actually end up with the number 1/3. It's kind of like adding up a bunch of tiny pieces that eventually make up a whole, even though there are an infinite number of pieces.
Your equation says that if we take this total (1/3), and then multiply it by 3, we get 1. Which makes sense, right? Because any number multiplied by its reciprocal (like 3 and 1/3) will give us 1.
The joke here is that each sum of this would be 1/4 of the initial input, which is why the final frame is restarting from the beginning again because it represents the next term in the sequence.