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. Thus, this provides another geometric interpretation for the series in Example 6.

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Timestamps

Exercise 2: Infinite sequence of circles – 0:00
Solution: Draw lines from the center of circles to the center of the big circle – 2:53
Obtain an equation involving the diameter of the small circle using the Pythagorean Identity – 7:20
Apply the difference of squares to find that the first small circle has a radius of 1/2 – 8:31
Similarly, we can find the diameter of the second circle – 12:19
Likewise, for the third circle, we notice a general pattern – 18:35
General formula for the diameter of small circles – 21:24
Calculating the second small circle’s diameter, which equals 1/6 – 22:38
The third small circle’s diameter is 1/12 – 24:05
Note that the general diameter terms follow the pattern 1/n * 1/(n+1) – 27:00
Proving the pattern using mathematical induction and partial fraction decomposition (from the earlier telescoping sum) – 28:12
Substituting our telescoping sum into the formula for the diameter, we obtain our proof – 30:23
The diameters of the smaller circles sum to 1, which is the radius of the bigger circle – 34:10