A Generalization to Cantor Series of Sondow’s Geometric Proof that e Is Irrational and His Measure of Its Irrationality

In 2006, Jonathan Sondow gave a nice geometric proof that e is irrational. Moreover, he said that a generalization of his construction may be used to prove the Cantor’s theorem. But, he didn’t do it in his paper, see [1]. So, this work will give a geometric proof to Cantor’s theorem using Sondow’s construction. After, it is given an irrationality measure to some Cantor series, for that, we generalize the Smarandache function. Finally, we give a irrationality measure for e that is a slight improvement the given one in [1].