Activity 8.3.3

For each of the following infinite geometric series, determine the values of $a$ and $r$, 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.

1. $1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots$

2. $4 - 2 + 1 - \frac{1}{2} + \frac{1}{4} - \cdots$

3. $2 + \frac{8}{3} + \frac{32}{9} + \frac{128}{27} + \cdots$

4. $\sum_{k = 0}^{\infty} 5 \cdot \left( \frac{3}{4} \right)^k$

5. $\sum_{k = 1}^{\infty} -2 \cdot \left( -\frac{2}{3} \right)^k$