One of my students was frustrated by the work involved in understanding infinite series. To spark his interest, I told him that the mysterious number that he has used most of his life, pi, is actually equivalent to an infinite series.
I told him that pi is equal to the series 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ... We decided to explore this series together. We used a calculator to find 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11. We found the this is approximately equal to 2.976. This number doesn't look very much like pi, so I suggested that we go further.
Next we used a calculator to find 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 + 4/17 - 4/19 + 4/21. We found that this is approximately equal to 3.232. This value is much closer to pi, and this time fell above pi, which is what we expected since we ended with a positive term.
As we continued to explore this sequence we found that the more terms we added, the closer our approximation got to pi. While this exercise didn't help the student solve problems involving geometric series, it was a small example of how fascinating math can be.