Use the geometric series formula 11y=n=0yn\displaystyle \frac{{1}}{{{1}-{y}}}={\sum_{{{n}={0}}}^{{\infty}}}{y}^{{n}} to express the function as a series:
x61+x=n=0\displaystyle \frac{{x}^{{6}}}{{{1}+{x}}}={\sum_{{{n}={0}}}^{{\infty}}}  \displaystyle