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 {Cn}. Find an expression for the diameter of Cn and thus provide another geometric demonstration of Example 7.

This is not a geometric series, so we go back to the definition of a convergent series and compute the partial sums.

We can simplify this expression if we use the partial fraction decomposition

Transcribed Image Text:

苘/(i + 1) 1.2 2.3 3.4 nin + 1)