Example  1
Determine the sum of each infinite geometric series, if a sum exists.

  1. \(10 + 5 + 2.5 + ...\)

    Step 1: Determine if the series is convergent or divergent.

    To determine if the sum of an infinite geometric series can be determined, you must first determine whether the geometric series is convergent or divergent; you must determine the value of \(r \).

    \[r = \frac{{t_n }}{{t_{n - 1} }} = \frac{5}{{10}} = \frac{1}{2} \]

    Step 2: If the series if convergent, determine the sum.

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

    \[\begin{align}
     S_\infty &= \frac{{t_1 }}{{1 - r}} \\
     S_\infty &= \frac{{10}}{{1 - \frac{1}{2}}} \\
     S_\infty &= 20 \\
     \end{align}\]

  2. \(1 + 1.5 + 2.25 + ...\)

    First, calculate \(r \).

    \[r = \frac{{t_n }}{{t_{n - 1} }} = \frac{{1.5}}{1} = 1.5 \]

    Because \(r > 1 \), this geometric series is divergent; therefore, a sum cannot be determined.
  3. \(100 - 98 + 96.04 - ...\)

    First, calculate \(r \).

    \[r = \frac{{t_n }}{{t_{n - 1} }} = \frac{{-98}}{{100}} = -0.98\]


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

    \[\begin{align}
     S_\infty &= \frac{{t_1 }}{{1 - r}} \\
     S_\infty &= \frac{{100}}{{1 - \left( {-0.98} \right)}} \\
     S_\infty &= 50.\overline {50}  \\
     \end{align}\]