proof writing - Prove that 1/2 + 1/4 + 1/8 ....... = 1

I've often heard that instead of adding up to a little less than one, 1/2 + 1/4 + 1/8... = 1. Is there any way to prove this using equations without using Sigma, or is it just an accepted fact? I need it without Sigma so I can explain it to my little sister.

It is not a duplicate because this one does not use Sigma, and the one marked as duplicate does. I want it to use variables and equations.

Answer

For physical intuition, so you can explain it to your little sister, I will use a 1m long ruler.

Take the ruler an divide it into two equal parts:

$$1=\frac{1}{2}+\frac{1}{2}$$

Take one of the parts you now have, and again divide it in half.

$$=\frac{1}{2}+\frac{1}{4}+\frac{1}{4}$$

Take one of the smaller parts you now have, and again divide it in half.

$$=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}$$

Repeat. In general for $n$ a positive integer,

$$=\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^n} \right)+\frac{1}{2^n}=1$$

So,

$$\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^n}=1-\frac{1}{2^n}$$

As we let $n$ become a really big (positive) integer, note the sum gets closer and closer to $1$, because $\frac{1}{2^n}$ gets really close to zero (the smallest part of the ruler you have left over gets close to 0 meters in length). We say the sum converges to $1$ in the limit that $n \to \infty$.