Write the sum in sigma notation $LaTeX: \displaystyle 6-\frac{13}{2}+\frac{169}{24}\ldots$ and then find the sum.

Taking the ratio of the first two terms gives $LaTeX: \displaystyle r = \frac{a_2}{a_1}=\frac{- \frac{13}{2}}{6}=- \frac{13}{12}$ . Using $LaTeX: \displaystyle a_{1}$ and $LaTeX: \displaystyle r$ to write in sigma notation gives $LaTeX: \displaystyle \sum_{n=1}^{\infty}6\left( - \frac{13}{12} \right)^{n-1}$ . Since $LaTeX: \displaystyle |r| = \frac{13}{12} \geq 1$ the geometric series diverges.