The Journal of Online Mathematics and Its Applications, Volume 7 (2007)
Bouncing Balls and Geometric Series, Robert Styer and Morgan Besson

Annotated Bibliography of Geometric Series Proofs

We have seen a geometric proof and a classic algebraic proof for the sum of the geometric series, but there are many more. Here we point the reader to some of these proofs. The first four deal with the finite geometric series, the rest with the infinite series. Does this suggest that the infinite case might be easier than the finite?

Our interactive proof was based on the visual proof by Klein and Bivens (reference below) which uses similar triangles. Other authors also use similar triangles to derive the sum of the infinite geometric series. Here is a list of some "similar triangle" proofs:

The wonderful books edited by Roger B. Nelsen and published by the Mathematical Association of America contain more beautiful visual demonstrations. From Proofs without Words,1993:

From Proofs without Words II, 2000:

Some other references on teaching geometric series:

Using Geometric Series Proofs in A Course

We have used this annotated bibliography of geometric series proofs in our sophomore Foundations of Mathematics course, taken by math majors. The students compare and contrast three or more proofs. In particular, we ask them to consider these questions:

  1. Does the proof apply to the finite geometric series or only the infinite series?
  2. Where does the proof use |r| < 1? Is there any way to interpret the proof if r > 1? What if −1 < r < 0?
  3. Where is the limit concept implicit in the proof? In the classic proof, we use the fact that rn → 0 as n → ∞. Is this implicit in this proof?
  4. Which proof is most elegant? Most understandable?