Write the sum in sigma notation LaTeX:  \displaystyle -6+\frac{21}{5}-\frac{147}{50}\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{21}{5}}{-6}=- \frac{7}{10} . Using LaTeX:  \displaystyle a_{1} and LaTeX:  \displaystyle r to write in sigma notation gives LaTeX:  \displaystyle \sum_{n=1}^{\infty}-6\left( - \frac{7}{10} \right)^{n-1} . Using the formula for the sum of an infinite geometric series gives LaTeX:  \displaystyle \frac{-6}{1-(- \frac{7}{10})}=- \frac{60}{17}