Chapter 8.2, Problem 47E

The figure shows two circles C and D of radius 1 that touch at P. T is a common tangent line; C₁ is the circle that touches C, D, and T; C₂ is the circle that touches C, D, and C₁; C₃ is the circle that touches C, D, and C₂. This procedure can be continued indefinitely and produces an infinite sequence of circles {C_n}. Find an expression for the diameter of C_n and thus provide another geometric demonstration of Example 6.

EXAMPLE 6 Show that the series ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n(n+1)}}$ is convergent, and find its sum.