Abstract
Before You Get Started. You probably know that the sequence of real numbers \(\frac{{n - 1}}{n}\) converges to the real number 1 as n gets larger. What exactly does this mean? Can you guess a definition of “converges” that does not use words like “nearer and nearer to” or “approaches”? Think about the picture above. Each black triangle occupies \(\frac{{1}}{4}\) of the area of its “parent” triangle. If the area of the biggest triangle is 1, then the sum of the areas of all the black triangles should be thought of as \(\frac{1}{4} + \frac{1}{{4^2 }} + \frac{1}{{4^3 }} \cdots.\) You might remember from previous math courses that the sum of this series is \(\frac{{1}}{3}\). Can you see in the picture that the blackened area constitutes \(\frac{{1}}{3}\) of the total area of the biggest white triangle? (Look at the white-black-white horizontal rows of triangles.) This is an example of convergence.
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