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Determine the sum of each infinite geometric series, if a sum exists.

  1. \(18 - 6 + 2 - ... \)

    First, decide if the infinite geometric series is convergent or divergent by calculating \(r \).
    \[r = \frac{{t_n }}{{t_{n - 1} }} = \frac{{-6}}{{18}} = -\frac{1}{3}\]

    Because \(-1 < r < 1 \), the infinite geometric series is convergent; therefore, a sum can be determined.

    \[\begin{align}
     S_\infty &= \frac{{t_1 }}{{1 - r}} \\
     S_\infty &= \frac{{18}}{{1 - \left( {-\frac{1}{3}} \right)}} \\
     S_\infty &= 13.5 \\
     \end{align}\]


  2. \(3 - 9 + 27 - ... \)

    \[r = \frac{{t_n }}{{t_{n - 1} }} = \frac{{-9}}{3} = -3\]

    Because \(r < -1 \), the infinite geometric series is divergent; therefore, a sum cannot be determined.

  3. \(60 + 6 + 0.6 + ... \)

    \[r = \frac{{t_n }}{{t_{n - 1} }} = \frac{6}{{60}} = 0.1\]

    Because \(-1 < r < 1 \), the infinite geometric series is convergent; therefore, a sum can be determined.
    \[\begin{align}
     S_\infty &= \frac{{t_1 }}{{1 - r}} \\
     S_\infty &= \frac{{60}}{{1 - 0.1}} \\
     S_\infty &= 66.\overline 6  \\
     \end{align} \]