## Proofs Without Words

Here are two of my favorite Proofs Without Words.  I’ve been thinking about infinite geometric series a lot lately, and these are two lovely, well-known, visualizations of two amazing infinite sums:

In a square of side length 1 (and therefore, area 1), cut the square in half; then cut one half in half (that’s a quarter); now cut one of the quarters in half (that’s an eighth); and so on and so on and so on (this puts the infinite in infinite sum).  Eventually you’ll fill up the whole squareSo this is a demonstration of the following amazing, and somewhat counterintuitive, fact that

Similarly, this diagram

is a visual representation of the following sum:

As any good, lazy mathematician would say, the details are left to the reader.

### 6 Comments

1. Rick says:

The sum below is equivalent to the square one:

3/4 + 3/16 + 3/64 + … =

(1/2 + 1/4) + (1/8 + 1/16) + (1/32 + 1/64) …

2. MrHonner says:

Cool! I wonder if that equivalence could be visualized in an elegant way.

• Taylor says:

“I wonder if that equivalence could be visualized in an elegant way.” Ever the teacher, Mr. Honner.

• MrHonner says:

I learned long ago that continually asking questions is a great way to deflect attention!

3. Quinn Culver says:

Hi,

I used one of these pictures here: http://math.stackexchange.com/questions/382295/prove-by-mathematical-induction-that-1-1-4-1-4n-4-3/382302#382302. I found it via a google search. Please let me know if that’s not okay.

Quinn