While the infinite geometric series has been explored from an algebraic perspective, is it possible to represent convergent series through a geometric picture? We have explored new ways of approaching the subject of infinite series to create a simple picture of their sums based on basic geometric techniques. Through the use of circle packing, the sum of an infinite series can be shown as circles cascade inward to infinity. Additionally, more complex convergent series can be represented through stereographic projection of points in the series onto a line or the unit circle. Though infinite series have been explored extensively through algebraic techniques, we want to create a new viewpoint on convergent series with a geometric approach. The new ways we have found to represent these series are both simple and intriguing. Our research will be presented through an oral presentation with an accompanying PowerPoint.