Exercise 2: Sum of diameters of Infinite Circles
In this video, I show that the sum of diameters of an infinite number of circles contacting between two big circles sums to the radius of the bigger circles, thus providing a geometric interpretation of the series with terms 1/(n(n + 1)). I first use the trigonometric identity to obtain an equation involving the diameter of the smaller circles, and then use mathematical induction (as well as my previous telescoping sum result) to show that the diameter of subsequent small circles is identical to the series terms.
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