Abstract

Geometric series with ratios  r such that  take the form  ∑ rⁱ=1 + r + r²  + r³  + ... = 1/(1-r)  when all the terms are added and  ∑ -rⁱ  = -1 - r - r²  - r³  - ... = -1/(1-r) when all the terms are subtracted. Our research examines the series  ∑ a_nrⁱ, where r > 0 and each a_n is either 1 or -1. The set of the series’ sums (sum set) has surprising structure.  To explore this structure, we separated the sets into three categories depending on the ratio. In these categories, we examined the effect of the sequences on the convergence of the series. Cantor sets make an appearance along with other exciting mathematical structures.