 In this context, attention has been focused on criteria sufficient to guarantee convergence of the trigonometric series almost everywhere (that is, for almost every value of the angle "θ" and of the distortion factor "ω").Greater insight into the underlying question that motivates the investigation of lacunary power series and lacunary trigonometric series can be gained by re-examining the simple example above.In that example we used the geometric series

and the Weierstrass M-test to demonstrate that the simple example defines an analytic function on the open unit disk.