Here is a

geometrical way to prove that the sum of an

infinite series can indeed be

finite:

What if you would take a square, add half of it onto itself, then add half of that, plus half of that, forever? Since you would keep adding all the time, even if the pieces being added grow smaller for every time, you would get an infinite amount right?

**Wrong.** Here's how to prove it:

Take a square. Next to said square, add half that square. Add half of that square below the second square. Proceed ad infinitum. Illustration:

1. -----------------
| |
| |
| |
| |
| |
| |
| |
| |
-----------------
2. ---------------------------------
| | |
| | |
| | |
| |_______________|
| |
| |
| |
| |
-----------------
3. ---------------------------------
| | | (I do wonder if
| | | images on E2
| | | would be such
| |_______________| a bad idea
| | | | after all.)
| | | |
| | | |
| | | |
----------------- ---------
4. ---------------------------------
| | |
| | |
| | |
| |_______________|
| | | |
| |_______| |
| | | |
| | | |
----------------- ---------
5. ---------------------------------
| | |
| | |
| | |
| |_______________|
| | | |
| |_______| |
| | | | |
| | | | |
----------------- -------------

Ok, you should've gotten the point by now. Anyway, as you can see, you can keep adding pieces as much as you like, but you'll never ever exceed the size of twice the original square. Thus, the infinite series will converge.