# Proof without words: Geometric series

1. Apr 29, 2012

### Bipolarity

On this website http://www.albany.edu/~bd445/Eco_466Y/Slides/Infinite_Geometric_Sum,_Proof_Without_Words.pdf [Broken] there is a "proof without words" of the sum of the infinite geometric series. However, I don't understand what makes the proof valid. In what order were the constructions done etc.? What were the arguments of the proof?

I am referring to the very first picture with the similar triangles.

BiP

Last edited by a moderator: May 5, 2017
2. Apr 29, 2012

### tiny-tim

Hi Bipolarity!

Triangle PQR is similar to triangle TSP.

So the two tans are the same.

3. Apr 29, 2012

### Bipolarity

I get that part, but I don't understand how you can construct the line that is equal to the geometric sum.

BiP

4. Apr 29, 2012

### tiny-tim

oh i see!

those diminishing quadrilaterals are similar to each other …

each has two sides the same, and one side r times those two

5. Apr 29, 2012

### Bipolarity

I don't understand how the quadrilaterals were constructed in the first place?

BiP

6. Apr 29, 2012

### tiny-tim

each quadrilateral is stuck onto the previous one, so they're each reduced by r

7. Apr 29, 2012

### Bipolarity

I get it now. Thanks.

BiP