Express the sum

 $\displaystyle \frac{{x}}{{2}}-\frac{{x}^{{2}}}{{4}}-\frac{{x}^{{3}}}{{8}}+\frac{{x}^{{4}}}{{16}}-\frac{{x}^{{5}}}{{32}}-\frac{{x}^{{6}}}{{64}}+\ldots$ 

 in terms of geometric series (hint:  group $\displaystyle {\left(\frac{{x}}{{2}}\right)}^{{{3}{n}}},{\left(\frac{{x}}{{2}}\right)}^{{{3}{n}-{1}}},$ and $\displaystyle {\left(\frac{{x}}{{2}}\right)}^{{{3}{n}-{2}}}$ ):

 $\displaystyle {\sum_{{{n}={0}}}^{{\infty}}}$ Preview Question 6 Part 1 of 2  

Now express the sum as a rational function:

Preview Question 6 Part 2 of 2