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# The figure shows two circles $C$ and $D$ of radius 1 that touch at $P.$ The line $T$ is a common tangent line; $C_1$ is the circle that touches $C, D,$ and $T; C_2$ is the circle that touches $C, D,$ and $C_1; C_3$ is the circle that touches $C, D,$ and $C_2.$ This procedure can be continued indefinitely and produces an infinite sequence of circles $\left \{C_n \right \}.$ Find an expression for the diameter of $C_n$ and thus provide another geometric demonstration of Example 8.

## $\sum_{n=1}^{\infty} a_{n}=\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=1,$ which is what we wanted to prove.

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##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

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