Home /
Expert Answers /
Advanced Math /
the-cantor-set-plays-an-important-role-in-modern-analysis-and-has-several-interesting-properties-it-pa707

The Cantor set plays an important role in modern analysis and has several interesting properties. It is constructed by starting with the interval [0, 1] (first line below) and removing the middle third; that is, take away the interval (1/3, 2/3). That leaves two segments: [0,1/3] and [2/3, I] (second line below). From each of those remove the middle third, the intervals (1/9,2/9) and (7/9, 8/9). This leaves four segments (third line below). From each of the four segments, remove the middle third. It can be shown that the total length of the intervals removed at each step is given by an = (2n-1) / (3n), n=1,2,... Find the total length of the intervals removed as this process continues forever (that is, as n > infinty).