Our first goal is to show students visual proofs for sums of specific geometric series. Show them one or two of the following and tell them which equation is represented. For the remaining images, ask the students what equations are being represented.

$$\dfrac{1}{2} + \dfrac{1}{2^2} + \dfrac{1}{2^3} + \ldots = 1$$

$$\dfrac{1}{3} + \dfrac{1}{3^2} + \dfrac{1}{3^3} + \ldots = \dfrac{1}{2}$$

$$\dfrac{1}{4} + \dfrac{1}{4^2} + \dfrac{1}{4^3} + \ldots = \dfrac{1}{3}$$

After knowing the above figure is 1/3, show the students the other 1/3 diagrams, here, and here. Seeing that the same equation may have multiple representations may prompt students to look for alternative representations for this and other geometric series.

Additional problems can be found here. Our next goal is to show students visual proofs for the generic infinite sum formula. Show your students the following image and ask them why this proves the infinite sum formula for geometric series:

Hopefully they found that \(TSP \sim PQR,\) from which it follows that

$$\begin{align} \dfrac{TS}{SP} &= \dfrac{PQ}{QR} \\[1em] 1 + r + r^2 + \ldots &= \dfrac{1}{1 - r} \end{align}$$ Next, reproduce this image and ask students why it proves the formula. The labels here are a bit blurry, but they read \(ar,\) \(ar^2,\) etc.For why this image proves the formula, you can watch this. A more formal, but slightly harder to follow explanation, can be found here. After students are satisfied with the visual proof, give them this explanation, which is easy to understand, assuming students already know how to find limits of rational functions. Next, give your students this challenge. Finally, give them this one.