Geometric Series: (½)ⁿ
This came from a math class argument. Someone said "you can add an infinite number of things together and get a finite answer" and I said that didn't make sense. Like, doesn't adding forever mean the sum keeps growing forever?
Turns out no, and this is why: each term is half the previous one. (1/2) + (1/4) + (1/8)... the pieces get small so fast that they never quite fill up the whole square, but they get infinitely close to it.
The slider adds one term at a time. Watch the remaining gray gap — it always cuts in half. By term 6 you've got 63/64 of the square filled. You'll never finish, but you can get as close as you want.
Fill the Square
The Infinite Sum That Equals 1
The formal answer is a/(1-r) where r is the ratio between terms. Here r = 1/2, so the sum is (1/2)/(1 - 1/2) = 1. That formula comes from the formula for geometric series, and once you understand it visually, it's obvious why it works.
This shows up in probability (the chance of eventually winning a game), in computing (binary fractions, like how 0.1111... in binary equals 1), and in calculus as the first example of a convergent infinite series. Every calculus class uses this series to build intuition for convergence before getting into the harder stuff.
I keep thinking about what "approaching but never reaching" actually means. The sum never equals 1 at a finite step. But the limit is exactly 1. That's the thing I'm still sitting with.