 The geometric series itself defines an analytic function that converges everywhere on the "closed" unit disk except when "z" = 1, where "g"("z") has a simple pole.And, since "z" = "e""iθ" for points on the unit circle, the geometric series becomes

at a particular "z", |"z"| = 1.From this perspective, then, mathematicians who investigate lacunary series are asking the question