Write the sum in sigma notation LaTeX: \displaystyle 4+\frac{72}{31}+\frac{1296}{961}\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{72}{31}}{4}=\frac{18}{31} . Using LaTeX: \displaystyle a_{1} and LaTeX: \displaystyle r to write in sigma notation gives LaTeX: \displaystyle \sum_{n=1}^{\infty}4\left( \frac{18}{31} \right)^{n-1} . Using the formula for the sum of an infinite geometric series gives LaTeX: \displaystyle \frac{4}{1-(\frac{18}{31})}=\frac{124}{13}