1. The geometric series $S={u}_{1}+{u}_{2}+{u}_{3}+\ldots+{u}_{n}+\ldots$ is convergent, and the sum to infinity of the series is ${S}_{\infty}$.  The first term of the series is $a$, and the common ratio is $r$.

A different series $T={u}_{1}^{2}+{u}_{2}^{2}+{u}_{3}^{2}+\ldots+{u}_{n}^{2}+\ldots$ is formed by squaring all the terms of the series $S$ above.

Show that $T$ is also a convergent geometric series.