Write the sum in sigma notation $LaTeX: \displaystyle -2+\frac{32}{3}-\frac{512}{9}\ldots$ and then find the sum.

Taking the ratio of the first two terms gives $LaTeX: \displaystyle r = \frac{a_2}{a_1}=\frac{\frac{32}{3}}{-2}=- \frac{16}{3}$ . Using $LaTeX: \displaystyle a_{1}$ and $LaTeX: \displaystyle r$ to write in sigma notation gives $LaTeX: \displaystyle \sum_{n=1}^{\infty}-2\left( - \frac{16}{3} \right)^{n-1}$ . Since $LaTeX: \displaystyle |r| = \frac{16}{3} \geq 1$ the geometric series diverges.