Activity 8.3.3.
For each of the following infinite geometric series, determine the values of \(a\) and \(r\text{,}\) compute the partial sums \(S_{5}\) and \(S_{10}\) exactly (writing each as a fraction), and if the infinite geometric series converges, find its value.
(a)
\(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots\)
(b)
\(4 - 2 + 1 - \frac{1}{2} + \frac{1}{4} - \cdots\)
(c)
\(2 + \frac{8}{3} + \frac{32}{9} + \frac{128}{27} + \cdots \)
(d)
\(\sum_{k = 0}^{\infty} 5 \cdot \left( \frac{3}{4} \right)^k\)
(e)
\(\sum_{k = 1}^{\infty} -2 \cdot \left( -\frac{2}{3} \right)^k\)
